Topological Types of Convergence for Nets of Multifunctions

1. Introduction

We denote the closure (resp. interior) of a subset A of a topological space $(X,\tau )$ by $Cl\left(A\right)$ (resp. $Int\left(A\right)$). Throughout the paper, it is assumed that $(X,\tau )$ is $T_1$. For any nonempty set A, let $\mathcal{P}\left(A\right)$ denote the family of all nonempty subsets of A. Generally, we will use the notation ${\mathcal{P}}^{n+1}\left(A\right)=\mathcal{P}\left({\mathcal{P}}^n\left(A\right)\right)$ where $n\in \mathbb{N}$ and ${\mathcal{P}}^0\left(A\right)=A$. The hyperspace topologies which we are about to use were introduced by L. Vietoris in [1], [2] (see also [3], [4]). We recall that for a topological space $(X,\tau )$, the upper (resp. lower) Vietoris topology ${\tau }^u$ (resp. ${\tau }^l$) on $\mathcal{P}\left(X\right)$ is generated by the basis $\left\{\left\{A\subset X:A\subset U\right\}:U\in \tau \right\}$ (resp. subbasis $\left\{\left\{A\subset X:A\cap U\ne \varnothing \right\}:U\in \tau \right\}$). The upper Vietoris topology plays a crucial role in our investigation. The closure of a subset $\mathcal{A}\subset \mathcal{P}\left(X\right)$ with respect to ${\tau }^u$ will be denoted by $\overline{\mathcal{A}}$.

The following specific property of the upper Vietoris topology will be used later, without explicitly referring to it.

Remark 1. 

For any family $\lambda \subset {\mathcal{P}}^2$(X) and U $\in \tau$ the following properties are equivalent:

• $\mathcal{P}\left(U\right)\cap {\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}\ne \varnothing$ and

• $\mathcal{P}\left(U\right)\cap \overline{\mathcal{K}}\ne \varnothing$ for all $\mathcal{K}\in \lambda$.

By a multifunction $F:(X,\tau )\to (Y,\sigma )$ from one topological space $(X,\tau )$ to another $(Y,\sigma )$ we mean a point-to-set correspondence and we always assume that $F\left(x\right)\ne \varnothing$ for all $x\in X$. Given a multifunction F, we denote its upper and lower inverse images of a subset B of Y by $F^{+}\left(B\right)$ and $F^{-}\left(B\right)$, respectively, i.e., $F^{+}\left(B\right)=\left\{x\in X:F\left(x\right)\subset B\right\}$ and $F^{-}\left(B\right)=\left\{x\in X:F\left(x\right)\cap B\ne \varnothing \right\}$ [5]. Of course, a single-valued function $f:X\to Y$ is naturally associated with the multifunction F given by $F\left(x\right)=\left\{f\left(x\right)\right\}$ for all $x\in X$. Obviously, in this case, $F^{+}\left(B\right)=F^{-}\left(B\right)=f^{-1}\left(B\right)$ and $F\left(A\right)=f\left(A\right)$ for all $A\subset X$ and $B\subset Y$. The inverse images of a multifunction F we formulate in the topological terms by using the following two functions from X to ${\mathcal{P}}^2\left(Y\right)$:

$x⟶\overline{\left\{F\left(x\right)\right\}}$ and $x⟶\overline{\mathcal{P}\left(F\right(x\left)\right)}$ for all $x\in X$.

The usefulness of these functions results from the following properties

Remark 2. 

For any multifunction $F:(X,\tau )⟶(Y,\sigma )$, $x\in X$ and $W\in \sigma$, the following equivalences hold:

• $x\in F^{+}\left(W\right)$ if and only if $\mathcal{P}\left(W\right)\cap \overline{\left\{F\left(x\right)\right\}}\ne \varnothing$ and,

• $x\in F^{-}\left(W\right)$ if and only if $\mathcal{P}\left(W\right)\cap \overline{\mathcal{P}\left(F\right(x\left)\right)}\ne \varnothing$.

It follows from the following obvious equivalences for any $B\subset Y$ and $W\in \sigma$:

• $B\subset W$ if and only if $\mathcal{P}\left(W\right)\cap \overline{\left\{B\right\}}\ne \varnothing$ and,

• $B\cap W\ne \varnothing$ if and only if $\mathcal{P}\left(W\right)\cap \overline{\mathcal{P}\left(B\right)}\ne \varnothing$.

In this paper, we will use the notion of filter as a tool for studying multifunctions. For the sake of fixing notation, we recall some classical notions that can be found, e.g., in [6] and [7]. A filter base $\mathcal{B}$ on a set Y is a nonempty collection of nonempty subsets of Y such that the intersection of any two members of $\mathcal{B}$ contains a member of $\mathcal{B}$. A filter $\mathcal{F}$ on a set Y is a nonempty collection of nonempty subsets of Y such that:

• if A$\in \mathcal{F}$ and A⊂B, then B$\in \mathcal{F}$ and

• if A$\in \mathcal{F}$ and B$\in \mathcal{F}$, then A∩B$\in \mathcal{F}$ for all $A,B\subset Y$.

For a family $\mathcal{A}$ of subsets of Y, let [$\mathcal{A}$] denote the family that consists of all supersets of each $A\in \mathcal{A}$. Every filter base $\mathcal{B}$ on Y generates a unique the smallest filter containing $\mathcal{B}$ as a subset, it consists of the subsets of Y containing a member of $\mathcal{B}$, i.e., it is equal to $\left[\mathcal{B}\right]$. Obviously, if $\mathcal{B}\subset \mathcal{A}\subset \left[\mathcal{B}\right]$, then $\left[\mathcal{A}\right]=\left[\mathcal{B}\right]$. A family of type $\left[\right\{B\left\}\right]$, where $B\subset Y$, is called the principal filter generated by B. In the case when $B=\left\{y\right\}$ for some $y\in Y$, the term principal filter generated by y is used.

G. Choquet [7] has used the concept of supremum (Sup) and the infimum (Inf) of filters on $\mathcal{P}\left(Y\right)$ as follows.

If $(Y,\sigma )$ is a topological space and $\lambda$ is a filter on $\mathcal{P}\left(Y\right)$, then

• the supremum of $\lambda$ is the set of all points $y\in Y$ such that for every open set W containing y and for each $\mathcal{K}\in \lambda$ there exists $K\in \mathcal{K}$ such that $W\cap$ K $\ne \varnothing$ and,

• the infimum of $\lambda$ is the set of all points $y\in Y$ such that for every open set W containing y, there exists $\mathcal{K}\in \lambda$ such that for each K$\in \mathcal{K}$, W∩ K $\ne \varnothing$.

In this paper, we will use Choquet’s concept of limits in the case of filters $\xi \subset {\mathcal{P}}^3\left(Y\right)$ on ${\mathcal{P}}^2\left(Y\right)$, considered in terms of the upper Vietoris topology ${\sigma }^u$ on $\mathcal{P}\left(Y\right)$. To simplify the notation, we denote the infimum and supremum operations by $\mathcal{I}$ and $\mathcal{S}$, respectively. The following characterizations of the infimum and supremum operations will serve as useful tools in our investigations. They follow immediately from the definition and the equivalences from Remark 1.

Remark 3. 

Let $(Y,\sigma )$ be a topological space and $\xi \subset {\mathcal{P}}^3\left(Y\right)$ be a filter on ${\mathcal{P}}^2\left(Y\right)$. Then

• $\mathcal{I}\left(\xi \right)=\overline{{\bigcup }_{\lambda \in \xi }{\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}}$ and,

• $\mathcal{S}\left(\xi \right)={\bigcap }_{\lambda \in \xi }\overline{{\bigcup }_{\mathcal{K}\in \lambda }\mathcal{K}}$.

2. The Monoid of Cluster Operators

This chapter introduces the notions of cluster functions and cluster operators that are the far generalizations of the concepts of cluster sets for multifunctions in the sense of [8] and earlier, for single-valued functions studied in [9]. For that purpose, we will use the supremum and infimum operations represented by the pair (l, u) of functions $u,l$:${\mathcal{P}}^3\left(Y\right)\to {\mathcal{P}}^2\left(Y\right)$ defined in terms of the upper Vietoris topology ${\sigma }^u$ on $\mathcal{P}\left(Y\right)$ as follows:

• $u\left(\lambda \right)={\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}$ and

• $l\left(\lambda \right)=\overline{{\bigcup }_{\mathcal{K}\in \lambda }\mathcal{K}}$ for any $\lambda \in {\mathcal{P}}^3\left(Y\right)$.

The functions u and l are highly related to each other, as we will show in the lemma below. First, let us consider the relation ≅ on ${\mathcal{P}}^2\left(Y\right)$ defined by the formula:

• $\mathcal{A}\cong \mathcal{B}$, if $\mathcal{A}\subset \bigcap \left\{\overline{\mathcal{P}\left(W\right)}:\mathcal{P}\left(W\right)\cap \mathcal{B}\ne \varnothing ,W\in \sigma \right\}$,

where $\mathcal{A},\mathcal{B}\in {\mathcal{P}}^2\left(Y\right)$.

It is easy to see that the relation ≅ is symmetric. Indeed, if $\mathcal{A}\cong \mathcal{B}$, $B\in \mathcal{B}$, $\mathcal{P}\left(V\right)\cap \mathcal{A}\ne \varnothing$ and $\mathcal{P}\left(W\right)\cap \mathcal{B}\ne \varnothing$, where $V,W\in \sigma$, then using the assumption that $\mathcal{A}\subset \overline{\mathcal{P}\left(W\right)}$ we obtain $\mathcal{P}\left(W\right)\cap \mathcal{P}\left(V\right)\ne \varnothing$ which proves that $\mathcal{B}\subset \overline{\mathcal{P}\left(V\right)}$ and consequently, that $\mathcal{B}\cong \mathcal{A}$.

For any set X, the relation ≅ induces a relation on ${\mathcal{P}}^2{\left(Y\right)}^X$, also denoted by ≅, which consists of all pairs $(\alpha ,\beta )\in {\mathcal{P}}^2{\left(Y\right)}^X\times {\mathcal{P}}^2{\left(Y\right)}^X$ that satisfy the following condition

• $\alpha \left(x\right)\cong \beta \left(x\right)$ for all $x\in X$.

The following property will be used later to prove one of the two main results of this work about the continuity of limits of nets of multifunctions.

Lemma 1. 

Let $(Y,\sigma )$ be a topological space and let X be a nonempty set. If α and β are functions from X into ${\mathcal{P}}^2\left(Y\right)$ such that $\alpha \cong \beta$, then $l\left(\alpha \right(A\left)\right)\cong u\left(\beta \right(A\left)\right)$ for any $A\subset X$.

Proof. 

Assuming that $\alpha \cong \beta$, we will show that $\overline{{\bigcup }_{x\in A}\alpha \left(x\right)}\cong {\bigcap }_{x\in A}\overline{\beta \left(x\right)}$. We, therefore, take two arbitrary sets $W,V\in \sigma$ such that $\mathcal{P}\left(W\right)\cap {\bigcap }_{x\in A}\overline{\beta \left(x\right)}\ne \varnothing$ and $\mathcal{P}\left(V\right)\cap \overline{{\bigcup }_{x\in A}\alpha \left(x\right)}\ne \varnothing$. Then there exists $x\in A$ such that $\mathcal{P}\left(W\right)\cap \beta \left(x\right)\ne \varnothing$ and $\mathcal{P}\left(V\right)\cap \alpha \left(x\right)\ne \varnothing$. Using the assumption we obtain $\alpha \left(x\right)\subset \overline{\mathcal{P}\left(W\right)}$ and consequently, $\mathcal{P}\left(V\right)\cap \mathcal{P}\left(W\right)\ne \varnothing$ which implies that $\overline{{\bigcup }_{x\in A}\alpha \left(x\right)}\subset \overline{\mathcal{P}\left(W\right)}$ and ends the proof. □

The following characterizations of the infimum and supremum operations will serve as useful tools in our investigations. They follow immediately from the definitions and Remark 3.

Lemma 2. 

Let $(Y,\sigma )$ be a topological space and let $\xi \subset {\mathcal{P}}^3\left(Y\right)$ be a filter on ${\mathcal{P}}^2\left(Y\right)$. Then

$\left(i\right)$

$\mathcal{I}\left(\xi \right)=\overline{{\bigcup }_{\lambda \in \xi }{\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}}=l\circ u\left(\xi \right)$ and

($ii)$

$\mathcal{S}\left(\xi \right)={\bigcap }_{\lambda \in \xi }\overline{{\bigcup }_{\mathcal{K}\in \lambda }\mathcal{K}}=u\circ l\left(\xi \right)$.

2.1. Limits of Filters in Terms of the Upper Vietoris Topology

The following elementary property will be used in the proof of Theorem 1, which will be stated later.

Lemma 3. 

Let $(Y,\sigma )$ be a topological space, $\mathcal{A}\in {\mathcal{P}}^2\left(Y\right)$, and let $\xi \subset {\mathcal{P}}^3\left(Y\right)$ be a filter on ${\mathcal{P}}^2\left(Y\right)$ such that $\mathcal{A}\in \bigcap \xi$, then $\mathcal{I}\left(\xi \right)\subset \overline{\mathcal{A}}\subset \mathcal{S}\left(\xi \right)$.

Proof. 

Since $\overline{\mathcal{A}}\in \{\overline{\mathcal{K}}:\mathcal{K}\in \lambda \}$ for every $\lambda \in \xi$, we have ${\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}\subset \overline{\mathcal{A}}\subset \overline{{\bigcup }_{\mathcal{K}\in \lambda }\mathcal{K}}$ for any $\lambda \in \xi$ and consequently, $\overline{{\bigcup }_{\lambda \in \xi }{\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}}\subset \overline{\mathcal{A}}\subset {\bigcap }_{\lambda \in \xi }\overline{{\bigcup }_{\mathcal{K}\in \lambda }\mathcal{K}}$ which, according to the above lemma, finishes the proof. □

Any filter base ${\mathcal{B}}^{\xi }$ that generates a filter $\xi$ on ${\mathcal{P}}^2\left(Y\right)$, can be used to characterize the value of the operations $\mathcal{I}$ and $\mathcal{S}$ by the combinations $l\circ u$ and $u\circ l$, respectively. Namely,

$l\circ u\left(\xi \right)=l\circ u({\mathcal{B}}^{\xi }$) and $u\circ l\left(\xi \right)=u\circ l\left({\mathcal{B}}^{\xi }\right)$.

This property immediately follows from the following one.

Lemma 4. 

Let $(Y,\sigma )$ be a topological space and ${\mathcal{B}}^{*}\subset {\mathcal{P}}^3\left(Y\right)$ a filter base on ${\mathcal{P}}^2\left(Y\right)$. Then the following equalities hold for any family $\mathcal{B}$ such that ${\mathcal{B}}^{*}\subset \mathcal{B}\subset \left[{\mathcal{B}}^{*}\right]$:

(i)

$u\circ l\left({\mathcal{B}}^{*}\right)=u\circ l\left(\mathcal{B}\right)$ and

(ii)

$l\circ u\left({\mathcal{B}}^{*}\right)=l\circ u\left(\mathcal{B}\right)$.

Proof. 

Since $\mathcal{B}\subset \left[{\mathcal{B}}^{*}\right]$, for every $\lambda \in \mathcal{B}$ there exists ${\lambda }^{★}\in {\mathcal{B}}^{★}$ such that ${\lambda }^{★}\subset \lambda$, which gives $\overline{{\bigcup }_{\mathcal{K}\in {\lambda }^{★}}\mathcal{K}}\subset \overline{{\bigcup }_{\mathcal{K}\in \lambda }\mathcal{K}}$ and ${\bigcap }_{\mathcal{K}\in {\lambda }^{★}}\overline{\mathcal{K}}\supset {\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}$ i.e., $l\left({\lambda }^{★}\right)\subset l\left(\lambda \right)$ and $u\left({\lambda }^{★}\right)\supset u\left(\lambda \right)$. So, in summary, we now know that for every $l\left(\lambda \right)\in l\left(\mathcal{B}\right)$ (resp. $u\left(\lambda \right)\in u\left(\mathcal{B}\right)$) there exists $l\left({\lambda }^{★}\right)\in l\left({\mathcal{B}}^{★}\right)$ (resp. $u{\left(\lambda \right)}^{★}\in u\left({\mathcal{B}}^{★}\right)$) such that $l\left({\lambda }^{★}\right)\subset l\left(\lambda \right)$ (resp. $u\left({\lambda }^{★}\right)\supset u\left(\lambda \right)$). Consequently,

${\bigcap }_{{\lambda }^{*}\in {\mathcal{B}}^{*}}\overline{l\left({\lambda }^{*}\right)}\subset {\bigcap }_{\lambda \in \mathcal{B}}\overline{l\left(\lambda \right)}$ and $\overline{{\bigcup }_{{\lambda }^{*}\in {\mathcal{B}}^{*}}u\left({\lambda }^{*}\right)}\supset \overline{{\bigcup }_{\lambda \in \mathcal{B}}u\left(\lambda \right)}$ i.e.,

$u\circ l\left({\mathcal{B}}^{*}\right)\subset u\circ l\left(\mathcal{B}\right)$ and $l\circ u\left({\mathcal{B}}^{*}\right)\supset l\circ u\left(\mathcal{B}\right)$. $(*)$

Let us now note that the inclusion ${\mathcal{B}}^{*}\subset \mathcal{B}$ implies the inclusions

$\left\{l\left({\lambda }^{*}\right):{\lambda }^{*}\in {\mathcal{B}}^{*}\right\}\subset \left\{l\left(\lambda \right):\lambda \in \mathcal{B}\right\}$ and $\left\{u\left({\lambda }^{*}\right):{\lambda }^{*}\in {\mathcal{B}}^{*}\right\}\subset \left\{u\left(\lambda \right):\lambda \in \mathcal{B}\right\}$.

Hence, ${\bigcap }_{{\lambda }^{*}\in {\mathcal{B}}^{*}}\overline{l\left({\lambda }^{*}\right)}\supset {\bigcap }_{\lambda \in \mathcal{B}}\overline{l\left(\lambda \right)}$ and $\overline{{\bigcup }_{{\lambda }^{*}\in {\mathcal{B}}^{*}}u\left({\lambda }^{*}\right)}\subset \overline{{\bigcup }_{\lambda \in \mathcal{B}}u\left(\lambda \right)}$ i.e.,

$u\circ l\left({\mathcal{B}}^{*}\right)\supset u\circ l\left(\mathcal{B}\right)$ and $l\circ u\left({\mathcal{B}}^{*}\right)\subset l\circ u\left(\mathcal{B}\right)$.

Thus, according to $(*)$, the proof is completed. □

Remark 4. 

Let us note that the following hold for any topological space $(Y,\sigma )$ and a filter ξ on ${\mathcal{P}}^2\left(Y\right)$:

$l\circ l\left(\xi \right)=\mathcal{P}\left(Y\right)$ and $u\circ u\left(\xi \right)=\left\{Y\right\}$.

However, it is easy to check that in the case of a filter base ξ on ${\mathcal{P}}^2\left(Y\right)$, the above equalities do not have to be true.

Indeed, if $B\in \mathcal{P}\left(Y\right)$ and $B\in \mathcal{P}\left(W\right)$, where $W\in \sigma$, then we can take $\lambda ={\mathcal{P}}^2\left(Y\right)$ that, of course, belongs to ξ and $\mathcal{P}\left(B\right)\in \lambda$. So, certainly, $\mathcal{P}\left(W\right)\cap \mathcal{P}\left(B\right)\ne \varnothing$, which proves that $B\in \overline{{\bigcup }_{\lambda \in \xi }{\bigcup }_{\mathcal{K}\in \lambda }\mathcal{K}}$ and consequently, $\mathcal{P}\left(Y\right)\subset l\circ l\left(\xi \right)$.

Now, suppose that $B\notin \left\{Y\right\}$, then $B\in \mathcal{P}\left(W\right)$, where $W=Y\setminus \left\{y\right\}\in \sigma$ for some $y\notin B$. If we take $\lambda ={\mathcal{P}}^2\left(Y\right)\in \xi$ and $\mathcal{K}=\left\{\right\{y\left\}\right\}\in \lambda$, then $P\left(W\right)\cap \mathcal{K}=\varnothing$. So, $B\notin {\bigcap }_{\lambda \in \xi }{\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}$, i.e. $B\notin u\circ u\left(\xi \right)$. Therefore, we have proved that $u\circ u\left(\xi \right)=\left\{Y\right\}$. On the other hand, it is clear that $Y\in \overline{\mathcal{K}}$ for every $\mathcal{K}\in {\mathcal{P}}^2\left(Y\right)$, so $Y\in {\bigcap }_{\lambda \in \xi }{\bigcap }_{\mathcal{K}\in \lambda }\overline{\mathcal{K}}$, which means that $\left\{Y\right\}\subset u\circ u\left(\xi \right)$, and finishes the proof of the equalities.

Now, let us consider a function Ψ from X to ${\mathcal{P}}^2\left(Y\right)$, where X is a topological space with a topology τ. Then, for any $x_0\in X$, the family $\tau \left(x_0\right)=\left\{U\in \tau :x_0\in U\right\}$, is a filter base on X. So, the family ${\mathcal{B}}_{x_0}=\mathrm{\Psi }\left(\tau \left(x_0\right)\right)=\{\{\mathrm{\Psi }\left(x\right):x\in U\}:U\in \tau \left(x_0\right)\}$ is a filter base on ${\mathcal{P}}^2\left(Y\right)$ and then, as we will show, the following equalities are true:

$l\circ l\left({\mathcal{B}}_{x_0}\right)=\overline{{\bigcup }_{x\in X}\mathrm{\Psi }\left(x\right)}$ and $u\circ u\left({\mathcal{B}}_{x_0}\right)={\bigcap }_{x\in X}\overline{\mathrm{\Psi }\left(x\right)}$.

Indeed, by definition, we have

$l\circ l\left({\mathcal{B}}_{x_0}\right)=\overline{{\bigcup }_{U\in \tau \left(x_0\right)}{\bigcup }_{x\in U}\mathrm{\Psi }\left(x\right)}$ and $u\circ u\left({\mathcal{B}}_{x_0}\right)={\bigcap }_{U\in \tau \left(x_0\right)}{\bigcap }_{x\in U}\overline{\mathrm{\Psi }\left(x\right)}$.

Since $X\in \tau \left(x_0\right)$, we conclude that

$l\circ l\left({\mathcal{B}}_{x_0}\right)\supset \overline{{\bigcup }_{x\in X}\mathrm{\Psi }\left(x\right)}$ and $u\circ u\left({\mathcal{B}}_{x_0}\right)\subset {\bigcap }_{x\in X}\overline{\mathrm{\Psi }\left(x\right)}$.

The opposite inclusions are evident because $U\subset X$ for all $U\in \tau \left(x_0\right)$.

Remark 5. 

It is easy to see that the formulas for the functions u and l permit the use of $\overline{\mathcal{K}}$ instead of $\mathcal{K}$ for each $\mathcal{K}\in \lambda$. So, the following equalities are true for any filter base ${\mathcal{B}}^{\xi }$ which generates a filter ξ:

$\mathcal{I}\left(\xi \right)=\mathcal{I}\left(\overline{\xi }\right)=\mathcal{I}\left(\overline{{\mathcal{B}}^{\xi }}\right)$ and $\mathcal{S}\left(\xi \right)=\mathcal{S}\left(\overline{\xi }\right)=\mathcal{S}\left(\overline{{\mathcal{B}}^{\xi }}\right)$,

where $\overline{\zeta }=\{\{\overline{\mathcal{K}}:\mathcal{K}\in \lambda \}:\lambda \in \zeta \}$ for $\zeta \subset {\mathcal{P}}^3\left(Y\right)$.

We will follow the tradition by denoting the principal filter on ${\mathcal{P}}^2\left(Y\right)$ generated by ${\lambda }^{*}\subset {\mathcal{P}}^2\left(Y\right)$, as $\left[{\lambda }^{*}\right]$ instead of $\left[\left\{{\lambda }^{*}\right\}\right]$, and denoting the principal filter on ${\mathcal{P}}^2\left(Y\right)$ generated by $\mathcal{A}\in {\mathcal{P}}^2\left(Y\right)$, as $\left[\mathcal{A}\right]$ instead of $\left[\right\{\left\{\mathcal{A}\right\}\left\}\right]$.

Let us note some facts concerning the filters of type $\left[{\mathcal{P}}^2\left(A\right)\right]$, where $A\subset Y$, that will prove useful for the investigation of functions in terms of filters. Namely, using the lemma stated below and Remark 2, one can note the following fact

Remark 6. 

For a multifunction $F:(X,\tau )\to (Y,\sigma )$, $x\in X$ and $W\in \sigma$, the following equivalences are true:

• $\mathcal{P}\left(W\right)\cap l\circ u\left(\left[{\mathcal{P}}^2\left(F\left(x\right)\right)\right]\right)\ne \varnothing$ if and only if $x\in F^{+}\left(W\right)$ and,

• $\mathcal{P}\left(W\right)\cap u\circ l\left(\left[{\mathcal{P}}^2\left(F\left(x\right)\right)\right]\right)\ne \varnothing$ if and only if $x\in F^{-}\left(W\right)$.

Lemma 5. 

For any topological space $(Y,\sigma )$, $A\subset Y$ and $\mathcal{A}\subset \mathcal{P}\left(Y\right)$, the following hold:

(i)

$l\circ u\left(\left[{\mathcal{P}}^2\left(A\right)\right]\right)=\overline{\left\{A\right\}}$ and $u\circ l\left(\left[{\mathcal{P}}^2\left(A\right)\right]\right)=\overline{\mathcal{P}\left(A\right)}$,

(ii)

$l\circ u\left(\left[\mathcal{A}\right]\right)=u\circ l\left(\left[\mathcal{A}\right]\right)=\overline{\mathcal{A}}$.

Proof. 

(i). According to Remark 4, we have

• $l\circ u\left(\left[\left\{{\mathcal{P}}^2\left(A\right)\right\}\right]\right)={\bigcap }_{\mathcal{K}\in {\mathcal{P}}^2\left(A\right)}\overline{\mathcal{K}}$ and

• $u\circ l\left(\left[\left\{{\mathcal{P}}^2\left(A\right)\right\}\right]\right)=\overline{{\bigcup }_{\mathcal{K}\in {\mathcal{P}}^2\left(A\right)}\mathcal{K}}$.

So, if $B\in l\circ u\left(\left[{\mathcal{P}}^2\left(A\right)\right]\right)$ and B$\in \mathcal{P}\left(W\right)$, where $W\in \sigma$, then for $\mathcal{K}=\left\{A\right\}$ we obtain $\mathcal{P}\left(W\right)\cap \mathcal{K}\ne \varnothing$ i.e., $A\in \mathcal{P}\left(W\right)$ which proves that B$\in \overline{\left\{A\right\}}$. Conversely, if B$\in \overline{\left\{A\right\}}$, $\mathcal{K}\in {\mathcal{P}}^2\left(A\right)$ and B$\in \mathcal{P}\left(W\right)$, where $W\in \sigma$, then $\mathcal{P}\left(A\right)\subset \mathcal{P}\left(W\right)$ and, since $\mathcal{K}\subset \mathcal{P}\left(A\right)$, we have $\mathcal{P}\left(W\right)\cap \mathcal{K}\ne \varnothing$ which proves that B $\in {\bigcap }_{\mathcal{K}\in {\mathcal{P}}^2\left(A\right)}\overline{\mathcal{K}}$.

Now, suppose that $B\in u\circ l\left(\left[{\mathcal{P}}^2\left(A\right)\right]\right)$ and B$\in \mathcal{P}\left(W\right)$, where $W\in \sigma$, then there exists $\mathcal{K}\subset \mathcal{P}\left(A\right)$ such that $\mathcal{P}\left(W\right)\cap \mathcal{K}\ne \varnothing$. Consequently, for some $K\in \mathcal{K}$, $K\in \mathcal{P}\left(A\right)\cap \mathcal{P}\left(W\right)$, so $\mathcal{P}\left(W\right)\cap \mathcal{P}\left(A\right)\ne \varnothing$, which proves that B$\in \overline{\mathcal{P}\left(A\right)}$. Conversely, let $B\in \overline{\mathcal{P}\left(A\right)}$ and $B\in \mathcal{P}\left(W\right)$, where $W\in \sigma$, then there exist $K\subset Y$ such that $K\in \mathcal{P}\left(W\right)\cap \mathcal{P}\left(A\right)$, hence we have $\mathcal{K}=\left\{K\right\}\in {\mathcal{P}}^2\left(A\right)$ such that $\mathcal{P}\left(W\right)\cap \mathcal{K}\ne \varnothing$ which proves that $B\in \overline{{\bigcup }_{\mathcal{K}\in {\mathcal{P}}^2\left(A\right)}\mathcal{K}}$.

Part (ii) follows immediately from Remark 5. □

Since each filter on a set Y belongs to the set ${\mathcal{P}}^2$(Y), it is convenient to denote by ${\mathcal{P}}_{\phi }^2\left(Y\right)$ the set of all such filters. If a set Y is itself a filter, we will use the following notation:

${\mathcal{P}}_{{\phi }^n}^{2n}\left(Y\right)={\mathcal{P}}_{\phi }^2\left({\mathcal{P}}_{{\phi }^{n-1}}^{2(n-1)}\left(Y\right)\right)$ for n = 1, 2,...,

where ${\mathcal{P}}_{{\phi }^0}^0\left(Y\right)=Y$ and ${\mathcal{P}}_{{\phi }^1}^2\left(Y\right)={\mathcal{P}}_{\phi }^2\left(Y\right)$.

It is clear that, according to Lemma 2 and Remark 4, the operations $\mathcal{I}=l\circ u$ and $\mathcal{S}=u\circ l$, as well as the compositions $u\circ u$ and $l\circ l$, might be thought to be the functions from ${\mathcal{P}}_{\phi }^2\left({\mathcal{P}}^2\left(Y\right)\right)$ to ${\mathcal{P}}^2\left(Y\right)$.

Following Schwarz [10], for each function f:X →Y, we will also denote by f the induced function that assigns to each nonempty subset A of X its image f(A)$\in \mathcal{P}\left(Y\right)$. In general, for $\alpha \subset {\mathcal{P}}^n\left(X\right)$ and $n\in N$, we have $f\left(\alpha \right)$ = $\left\{f\left(\beta \right)\in {\mathcal{P}}^n\left(Y\right):\beta \in \alpha \right\}$. So, we will consider the following compositions:

${\mathcal{P}}_{{\phi }^n}^{2n}({\mathcal{P}}^2$(Y))$\stackrel{f_1}{⟶}{\mathcal{P}}_{{\phi }^{n-1}}^{2(n-1)}({\mathcal{P}}^2$(Y))$\stackrel{f_2}{⟶}$...$\stackrel{f_{n-1}}{⟶}{\mathcal{P}}_{\phi }^2({\mathcal{P}}^2$(Y))$\stackrel{f_n}{⟶}{\mathcal{P}}^2$(Y) i.e.,

$f_i:{\mathcal{P}}_{{\phi }^{n+1-i}}^{2(n+1-i)}({\mathcal{P}}^2$(Y))$⟶{\mathcal{P}}_{{\phi }^{n-i}}^{2(n-i)}({\mathcal{P}}^2$(Y)),

where i = 1, 2,..., n, and $f_i\in \left\{l\circ u,u\circ l,u\circ u,l\circ l\right\}$.

Equivalently,

$h_{2n}\circ h_{2n-1}\circ ...\circ h_2\circ h_1:{\mathcal{P}}_{{\phi }^n}^{2n}\left({\mathcal{P}}^2\left(Y\right)\right)⟶{\mathcal{P}}^2$(Y),

where $h_{2i}\circ h_{2i-1}:{\mathcal{P}}_{{\phi }^{n+1-i}}^{2(n+1-i)}({\mathcal{P}}^2$(Y))$⟶{\mathcal{P}}_{{\phi }^{n-i}}^{2(n-i)}({\mathcal{P}}^2$(Y)),

$i=1,2,...,n$ and ($h_{2i},h_{2i-1}$) $\in \left\{(l,u),(u,l),(u,u),(l,l)\right\}$.

Bearing in mind the case described in Remark 4 and according to Lemma 4, one can use in the above compositions instead of filters, their filter bases, when ($h_{2i},h_{2i-1}$) $\in \left\{(l,u),(u,l)\right\}$.

2.2. Cluster Operators

Given a topological space $(X,\tau )$, we will use the notation $\tau$ also for the function $\tau :X⟶{\mathcal{P}}^2\left(X\right)$ defined by the formula

$\tau \left(x\right)$ = $\left\{U\in \tau :x\in U\right\}$ for all $x\in X$.

We denote by ${\tau }^2$ the function ${\tau }^2:X⟶{\mathcal{P}}^4\left(X\right)$ defined as

${\tau }^2\left(x\right)=\tau \left(\tau \left(x\right)\right)$ for all $x\in X$, i.e.,

${\tau }^2\left(x\right)=\left\{\left\{\tau \left(u\right):u\in U\right\}:U\in \tau \left(x\right)\right\}$ for all $x\in X$.

In general, for any $n\in N$, we consider the function ${\tau }^n:X⟶{\mathcal{P}}^{2n}\left(X\right)$ defined by the pattern

${\tau }^n\left(x\right)=\tau ({\tau }^{n-1}$(x)) for all $x\in X$,

where ${\tau }^1=\tau$, and ${\tau }^0$ denotes the identity function on X.

It is easy to show inductively that ${\tau }^n\left({\tau }^k\left(x\right)\right)={\tau }^k\left({\tau }^n\left(x\right)\right)$ for all $n,k\in N$ and $x\in X$. Of course, because of the assumption that $(X,\tau )$ is a $T_1$ space, the function $\tau$ is injective.

Indeed, if $a,b\in X$ and $a\ne b$, then the subset $U=X\setminus \left\{a\right\}$ belongs to $\tau$ and $b\in U$, so $U\in \tau \left(b\right)$, but $U\notin \tau \left(a\right)$.

Remark 7. 

From now on, we will use the following three alternative formulas for ${\tau }^n\left(x_0\right)$, where $x_0\in X$ and $n=1,2,...$.

Directly from definition:

$\left\{...\left\{\left\{x_n:{\tau }^i\left(x_{n-i}\right)\in {\tau }^i\left(U_{n-i}\right)\right\}:{\tau }^i\left(U_{n-i}\right)\in {\tau }^{i+1}\left(x_{n-(i+1)}\right)\right\}:i=0,...,n-1\right\}$

or, on the short form:

$\left\{...\left\{x_n:{\mathcal{B}}_{(i-1)mod2}^{(n,⌊{\displaystyle \frac{i-1}{2}}⌋)}\in {\mathcal{B}}_{imod2}^{(n,⌊{\displaystyle \frac{i}{2}}⌋)}\right\}:i=1,...,2n\right\}$, where

${\mathcal{B}}_0^{(n,i)}$ = ${\tau }^i\left(x_{n-i}\right)$ for $i\in \left\{0,...,n\right\}$ and,

${\mathcal{B}}_1^{(n,i)}$ = ${\tau }^i\left(U_{n-i}\right)$ for $i\in \left\{0,...,n-1\right\}$.

So, any 2n-tuple of index sets of ${\tau }^n\left(x_0\right)$ is of the form

$({\mathcal{B}}_1^{(n,0)}$, ${\mathcal{B}}_0^{(n,1)}$, ${\mathcal{B}}_1^{(n,1)}$, ${\mathcal{B}}_0^{(n,2)}$, ${\mathcal{B}}_1^{(n,2)}$,..., ${\mathcal{B}}_1^{(n,n-1)}$, ${\mathcal{B}}_0^{(n,n)})\in$

${\mathcal{P}}^1\left(X\right)\times {\mathcal{P}}^2\left(X\right)\times ...\times {\mathcal{P}}^{2n-1}\left(X\right)\times {\mathcal{P}}^{2n}\left(X\right)$,

i.e., ${\mathcal{B}}_{kmod2}^{(n,⌊{\displaystyle \frac{k}{2}}⌋)}\in {\mathcal{P}}^k\left(X\right)$, where $k\in \left\{1,2,...,2n\right\}$.

The justification of the third formula comes from the lemma below.

$\left(iii\right)$$\left\{...\left\{\left\{x_n:z_{n-i}\in U_{n-i}\right\}:U_{n-i}\in \tau \left(x_{n-(i+1)}\right)\right\}:i=0,...,n-1\right\}$.

For the same reasons, we have the following two equivalences:

$\left(iv\right)$ (1) ${\mathcal{B}}_0^{(n,i)}\in {\mathcal{B}}_1^{(n,i)}$ if and only if ${\mathcal{B}}_0^{(n-m,i-m)}\in {\mathcal{B}}_1^{(n-m,i-m)}$ and,

(2) ${\mathcal{B}}_1^{(n,i)}\in {\mathcal{B}}_0^{(n,i+1)}$ if and only if ${\mathcal{B}}_1^{(n-m,i-m)}\in {\mathcal{B}}_0^{(n-m,i+1-m)}$

for $i\in \left\{0,...,n-1\right\}$ and $m\in \left\{0,...,i\right\}$.

Lemma 6. 

Let $(X,\tau )$ be a topological space, $x\in X$ and $i=1,2,...$ Then the following statements are equivalent for any $W\in \tau$:

$\left(i\right)$$x\in$W,

$\left(ii\right)$${\tau }^i\left(x\right)\in {\tau }^i\left(W\right)$,

$\left(iii\right)$${\tau }^i\left(W\right)\in {\tau }^{i+1}\left(x\right)$.

Proof. 

Let us first show that for any $n\in N$, the function ${\tau }^n$ is injective. If ${\tau }^2\left(a\right)={\tau }^2\left(b\right)$ for some $a,b\in X$, which means that $\left\{\tau \left(U\right):U\in \tau \left(a\right)\right\}=\left\{\tau \left(V\right):V\in \tau \left(b\right)\right\}$, then for every $U\in \tau \left(a\right)$ there exists $V\in \tau \left(b\right)$ such that $\tau \left(U\right)=\tau \left(V\right)$ which implies that $U=V$, because the function $\tau$ is injective. Consequently, $\tau \left(a\right)\subset \tau \left(b\right)$. Analogously, one can show that $\tau \left(b\right)\subset \tau \left(a\right)$. Since $\tau$ is injective, the equality $\tau \left(a\right)=\tau \left(b\right)$ implies that $a=b$ and proves that the function ${\tau }^2$ is injective. Now, assume that the function ${\tau }^k$ is injective for $k\in N$, and let ${\tau }^{k+1}\left(a\right)={\tau }^{k+1}\left(b\right)$ for $a,b\in X$. Then we have ${\tau }^k\left(\tau \left(a\right)\right)={\tau }^k\left(\tau \left(b\right)\right)$, so $\left\{{\tau }^k\left(U\right):U\in \tau \left(a\right)\right\}=\left\{{\tau }^k\left(V\right):V\in \tau \left(b\right)\right\}$ and, using the assumption on ${\tau }^k$, analogously as in the case of ${\tau }^2$, we obtain that $a=b$.

The implication $\left(i\right)\Rightarrow \left(ii\right)$ is obvious, so assume that ${\tau }^i\left(x\right)\in {\tau }^i\left(W\right)$. Then there exists a w $\in W$ such that ${\tau }^i\left(x\right)={\tau }^i\left(w\right)$ and, because of the injectivity property of ${\tau }^i$ we have $x=w$. So, $W\in \tau \left(x\right)$ and thus ${\tau }^i\left(W\right)\in {\tau }^{i+1}\left(x\right)$, i.e., $\left(iii\right)$. The assumption (iii) means that ${\tau }^i\left(W\right)\in {\tau }^i\left({\tau }^i\left(W\right)\right)=\left\{{\tau }^i\left(U\right):U\in \tau \left(x\right)\right\}$. Therefore, ${\tau }^i\left(W\right)={\tau }^i\left(U\right)$ for some $U\in \tau \left(z\right)$ and consequently $W=U$, so $x\in W$ which finishes the proof. □

It is evident that for a given pair $\left(\right(X,\tau )$, $(Y,\sigma ))$ of topological spaces, for any function $\mathrm{\Psi }:X⟶{\mathcal{P}}^2\left(Y\right)$ and $x\in X$, the set $\mathrm{\Psi }\circ \tau \left(x\right)=\left\{\left\{\mathrm{\Psi }\left(x_1\right):x_1\in U_1\right\}:U_1\in \tau \left(x\right)\right\}$ is a filter base on ${\mathcal{P}}^2\left(Y\right)$, so $[\mathrm{\Psi }\circ \tau \left(x\right)]\in {\mathcal{P}}_{\phi }^2\left({\mathcal{P}}^2\left(Y\right)\right)$. For the same reason, in the case of a function $\mathcal{F}:X⟶{\mathcal{P}}_{\phi }^2\left({\mathcal{P}}^2\left(Y\right)\right)$ we have $[\mathcal{F}\circ \tau \left(x\right)]\in {\mathcal{P}}_{{\phi }^2}^4\left({\mathcal{P}}^2\left(Y\right)\right)$.

In general, given a function $\mathrm{\Psi }:X⟶{\mathcal{P}}^2\left(Y\right)$ and $n\in \left\{1,2,3,...\right\}$, we define the function

$[\mathrm{\Psi }\circ {\tau }^n]:X⟶{\mathcal{P}}_{{\phi }^n}^{2n}\left({\mathcal{P}}^2\left(Y\right)\right)$ by

$[\mathrm{\Psi }\circ {\tau }^n]\left(x\right)$ = $\left[[\mathrm{\Psi }\circ {\tau }^{n-1}]\left(\tau \left(x\right)\right)\right]$ for all $x\in X$,

where $[\mathrm{\Psi }\circ {\tau }^1]\left(x\right)=[\mathrm{\Psi }\circ \tau \left(x\right)]$.

We also consider such a function corresponding to the number n = 0, defined by

$[\mathrm{\Psi }\circ {\tau }^0]\left(x\right)$ = $\left[\mathrm{\Psi }\right(x\left)\right]$ for all $x\in X$.

Of course, sets of the form $[\mathrm{\Psi }\circ {\tau }^n]\left(x\right)$, $x\in X$, $n\in \left\{1,2,...\right\}$, are arguments of the compositions $h_{2n}\circ h_{2n-1}\circ ...\circ h_2\circ h_1$, where $h_i\in \left\{u,l\right\}$ for i = 1, 2,..., 2n, so we can consider the following functions:

$h_{2n}\circ h_{2n-1}\circ ...h_2\circ h_1\circ [\mathrm{\Psi }\circ {\pi }^n]:X⟶{\mathcal{P}}^2\left(Y\right)$.

Because the number of the functions $h_i\in \left\{u,l\right\}$ for i = 1, 2,..., 2n, that is used in this expression is determined by ${\tau }^n$, we will use the following short notation to describe those compositions:

(C.F) $〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉:X\to {\mathcal{P}}^2\left(Y\right)$.

The function $[\mathrm{\Psi }\circ {\pi }^0]$ plays a special role since, according to Lemma 5(ii), we have $l\circ u\circ [\mathrm{\Psi }\circ {\tau }^0]\left(x\right)=u\circ l\circ [\mathrm{\Psi }\circ {\tau }^0]\left(x\right)=\overline{\mathrm{\Psi }\left(x\right)}$ for all $x\in X$. So, denoting the function $l\circ u\circ [\mathrm{\Psi }\circ {\tau }^0]=u\circ l\circ [\mathrm{\Psi }\circ {\pi }^0]$ by $\overline{\mathrm{\Psi }}$, i.e.,

$\overline{\mathrm{\Psi }}\left(x\right)=\overline{\mathrm{\Psi }\left(x\right)}$ for all $x\in X$,

and bearing in mind Remark 5, we obtain the following equality:

$h_{2n}\circ h_{2n-1}\circ ...h_2\circ h_1\circ [\mathrm{\Psi }\circ {\tau }^n]=h_{2n}\circ h_{2n-1}\circ ...h_2\circ h_1\circ [\overline{\mathrm{\Psi }}\circ {\tau }^n]$

for all $n\in \left\{1,2,...\right\}$, where $h_i\in \left\{u,l\right\}$ for i = 1, 2,..., 2n.

Therefore, in the special case when n = 0, we mean that $\overline{\mathrm{\Psi }\left(x\right)}$ is the value of the function denoted in accordance with the above convention (CL.F), by $〈\mathrm{\Psi }〉$, i.e.,

$〈\mathrm{\Psi }〉=\overline{\mathrm{\Psi }}$.

So, according to the above findings, we have the following equality:

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉=〈h_{2n},h_{2n-1},...,h_2,h_1,〈\mathrm{\Psi }〉〉$

for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$.

Given $\mathcal{A}\in {\mathcal{P}}^2\left(Y\right)$, we denote by $〈\mathcal{A}〉$ the constant function taking the value $\overline{\mathcal{A}}$ i.e.,

$〈\mathcal{A}〉\left(x\right)=\overline{\mathcal{A}}$ for all $x\in X$.

For any composition $h_{2n}\circ h_{2n-1}\circ ...\circ h_2\circ h_1$, where $h_i\in \left\{u,l\right\}$ for i = 1, 2,..., 2n, we will use the symbol $〈h_{2n},h_{2n-1},...,h_2,h_1〉$ to denote the function

(C.O) $〈h_{2n},h_{2n-1},...,h_2,h_1〉:{\mathcal{P}}^2{\left(Y\right)}^X⟶{\mathcal{P}}^2{\left(Y\right)}^X$

defined by

$〈h_{2n},h_{2n-1},...,h_2,h_1〉\left(\mathrm{\Psi }\right)=〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉$

for all $\mathrm{\Psi }$:X$\to {\mathcal{P}}^2\left(Y\right)$.

For the case where $n=0$, $〈\mathrm{\Psi }〉$ is understood to be the value of the function of type (C.O) denoted by $〈...〉$, for the argument $\mathrm{\Psi }$, i.e.,

$〈...〉\left(\mathrm{\Psi }\right)=〈\mathrm{\Psi }〉$ for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$.

For a convenient terminology, we establish the following definition.

Definition 1. 

Let $(X,\tau )$ and $(Y,\sigma )$ be topological spaces and let Ψ be a function from X to ${\mathcal{P}}^2{\left(Y\right)}^X$.

(i) Any function of the form

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉:X\to {\mathcal{P}}^2\left(Y\right)$

defined in (C.F) will be called a cluster function.

(ii) By a cluster operator, we mean any function of the form

$〈h_{2n},h_{2n-1},...,h_2,h_1〉:{\mathcal{P}}^2{\left(Y\right)}^X⟶{\mathcal{P}}^2{\left(Y\right)}^X$

defined in (C.O).

The following simple properties will be used later, where $\mathcal{C}.\mathcal{O}(X,Y)$ denotes the collection of all cluster operators for a given pair $\left(\right(X,\tau ),(Y,\sigma \left)\right)$ of topological spaces.

Lemma 7. 

For any functions Ψ, ${\mathrm{\Psi }}^{*}\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $\mathcal{A}\in {\mathcal{P}}^2\left(Y\right)$, the following conditions hold:

$\left(i\right)$$〈h_{2n},h_{2n-1},...,h_2,h_1,〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉〉$=

$〈h_{2n},h_{2n-1},...,h_2,h_1,g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉$ for all

$〈h_{2n},h_{2n-1},...,h_2,h_1〉$, $〈g_{2m},g_{2m-1},...,g_2,g_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$.

$\left(ii\right)$ For any $〈h_{2n},h_{2n-1},...,h_2,h_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$ we have:

$\left(a\right)$$〈l,l,h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉=〈l,l,\mathrm{\Psi }〉$,

$\left(b\right)$$〈u,u,h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉=〈u,u,\mathrm{\Psi }〉$ and

$\left(c\right)$$〈l,u,〈\mathcal{A}〉〉=〈u,l,〈\mathcal{A}〉〉=〈\mathcal{A}〉$.

$\left(iii\right)$ For any $\alpha \in \left\{l,u\right\}$ we have:

$\left(a\right)$$〈u,l,u,\alpha 〉$ = $〈u,\alpha 〉$,

$\left(b\right)$$〈l,u,l,\alpha 〉$ = $〈l,\alpha 〉$,

$\left(c\right)$$〈\alpha ,u,l,u〉$ = $〈\alpha ,u〉$ and

$\left(d\right)$$〈\alpha ,l,u,l〉$ = $〈\alpha ,l〉$.

$\left(iv\right)$ If $\mathrm{\Psi }\left(x\right)\subset {\mathrm{\Psi }}^{*}\left(x\right)$ for all $x\in X$, then

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉\left(x\right)\subset 〈h_{2n},h_{2n-1},...,h_2,h_1,{\mathrm{\Psi }}^{*}〉\left(x\right)$ for all

$〈h_{2n},h_{2n-1},...,h_2,h_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$ and $x\in X$.

Proof. 

To show (i), let us observe that $g_{2m}\circ g_{2m-1}\circ ...\circ g_2\circ g_1\circ [\mathrm{\Psi }\circ {\tau }^{m+n}]=$

$[g_{2m}\circ g_{2m-1}\circ ...\circ g_2\circ g_1\circ [\mathrm{\Psi }\circ {\tau }^m]\circ {\tau }^n]=$

$[〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉\circ {\tau }^n]$, thus

$〈h_{2n},h_{2n-1},...,h_2,h_1,g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉=$

$h_{2n}\circ h_{2n-1}\circ ...\circ h_2\circ h_1\circ g_{2m}\circ g_{2m-1}\circ ...\circ g_2\circ g_1\circ [\mathrm{\Psi }\circ {\tau }^{m+n}]=$

$h_{2n}\circ h_{2n-1}\circ ...\circ h_2\circ h_1\circ [〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉\circ {\tau }^n]=$

$〈h_{2n},h_{2n-1},...,h_2,h_1,〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉〉$.

The properties (a) and (b) described in part (ii) follow immediately from Remark 4 after using (i). To check property (c), one needs only note that $[〈\mathcal{A}〉\circ {\pi }^1]\left(z\right)=\left[\overline{\mathcal{A}}\right]$ for any $z\in Z$, and then to use Lemma 5(ii) and Remark 5.

For the proof of (iii), we consider the two cases, $\alpha =u$ or $\alpha =l$, and then, using (ii), part (c) (resp. part (b)), we obtain $〈u,l,u,u〉$ = $〈u,u〉$ (resp. $〈u,u,l,u〉$ = $〈u,u〉$) or, using (ii), part (c) (resp. part (a)), we obtain $〈l,u,l,l〉$ = $〈l,l〉$ (resp. $〈l,l,u,l〉$ = $〈l,l〉$). So, it remains to show that $〈u,l,u,l〉$ = $〈u,l〉$ and $〈l,u,l,u〉$ = $〈l,u〉$.

For this purpose let us take a function $\mathrm{\Psi }:X\to {\mathcal{P}}^2\left(Y\right)$, $x\in X$, $U_1\in \tau \left(x\right)$ and let us apply Lemma 3. It is clear that for all $x_1\in U_1$ we have

$l\left(\mathrm{\Psi }\left(U_1\right)\right)\in \left\{l\left(\mathrm{\Psi }\left(U_2\right)\right):U_2\in \tau \left(x_1\right)\right\}$ and $u\left(\mathrm{\Psi }\left(U_1\right)\right)\in \left\{u\left(\mathrm{\Psi }\left(U_2\right)\right):U_2\in \tau \left(x_1\right)\right\}$, hence

${\bigcap }_{U_2\in \tau \left(x_1\right)}l\left(\mathrm{\Psi }\left(U_2\right)\right)\subset l\left(\mathrm{\Psi }\left(U_1\right)\right)$ and $u\left(\mathrm{\Psi }\left(U_1\right)\right)\subset \overline{{\bigcup }_{U_2\in \pi \left(x_1\right)}u\left(\mathrm{\Psi }\left(U_2\right)\right)}$.

Consequently,

$\overline{{\bigcup }_{x_1\in U_1}{\bigcap }_{U_2\in \tau \left(x_1\right)}l\left(\mathrm{\Psi }\left(U_2\right)\right)}\subset l\left(\mathrm{\Psi }\left(U_1\right)\right)$ and

$u\left(\mathrm{\Psi }\left(U_1\right)\right)\subset {\bigcap }_{x_1\in U_1}\overline{{\bigcup }_{U_2\in \tau \left(x_1\right)}u\left(\mathrm{\Psi }\left(U_2\right)\right)}$ for all $U_1\in \tau \left(x\right)$ which implies that

${\bigcap }_{U_1\in \tau \left(x\right)}\overline{{\bigcup }_{x_1\in U_1}{\bigcap }_{U_2\in \tau \left(x_1\right)}l\left(\mathrm{\Psi }\left(U_2\right)\right)}\subset {\bigcap }_{U_1\in \tau \left(x\right)}l\left(\mathrm{\Psi }\left(U_1\right)\right)$ and

$\overline{{\bigcup }_{U_1\in \tau \left(x\right)}u\left(\mathrm{\Psi }\left(U_1\right)\right)}\subset \overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}\overline{{\bigcup }_{U_2\in \tau \left(x_1\right)}u\left(\mathrm{\Psi }\left(U_2\right)\right)}}$, and proves the following inclusions:

• $〈u,l,u,l,\mathrm{\Psi }〉\left(x\right)\subset 〈u,l,\mathrm{\Psi }〉\left(x\right)$ and $〈l,u,\mathrm{\Psi }〉\left(x\right)\subset 〈l,u,l,u,\mathrm{\Psi }〉\left(x\right)$, respectively, for all $\mathrm{\Psi }:X\to {\mathcal{P}}^2\left(Y\right)$ and $x\in X.$

To prove the inverse inclusions, let us note that

$〈u,l,\mathrm{\Psi }〉\left(x\right)\in 〈u,l,\mathrm{\Psi }〉\left(U_1\right)$ and $〈l,u,\mathrm{\Psi }〉\left(x\right)\in 〈l,u,\mathrm{\Psi }〉\left(U_1\right)$ for all $U_1\in \tau \left(x\right)$.

So, $〈u,l,\mathrm{\Psi }〉\left(x\right)\in {\bigcap }_{U_1\in \tau \left(x\right)}〈u,l,\mathrm{\Psi }〉\left(U_1\right)\subset \bigcap [〈u,l,\mathrm{\Psi }〉\circ \tau \left(x\right)]$ and

$〈l,u,\mathrm{\Psi }〉\left(x\right)\in {\bigcap }_{U_1\in \tau \left(x\right)}〈l,u,\mathrm{\Psi }〉\left(U_1\right)\subset \bigcap [〈l,u,\mathrm{\Psi }〉\circ \tau \left(x\right)]$ which implies, according to Lemma 3, that

$〈u,l,\mathrm{\Psi }〉\left(x\right)\subset u\circ l\circ [〈u,l,\mathrm{\Psi }〉\circ \tau \left(x\right)]=〈u,l,〈u,l,\mathrm{\Psi }〉〉\left(x\right)=〈u,l,u,l,\mathrm{\Psi }〉\left(x\right)$ and

$〈l,u,l,u,\mathrm{\Psi }〉\left(x\right)=〈l,u,〈l,u,\mathrm{\Psi }〉〉\left(x\right)=l\circ u\circ [〈l,u,\mathrm{\Psi }〉\circ \tau \left(x\right)]\subset 〈l,u,\mathrm{\Psi }〉\left(x\right)$ So, the following inclusions hold true

$••$$〈u,l,\mathrm{\Psi }〉\left(x\right)\subset 〈u,l,u,l,\mathrm{\Psi }〉\left(x\right)$ and $〈l,u,l,u,\mathrm{\Psi }〉\left(x\right)\subset 〈l,u,\mathrm{\Psi }〉\left(x\right)$, respectively.

Finally, using • and $••$ we obtain $〈u,l,u,l〉$ = $〈u,l〉$ and $〈l,u,l,u〉$ = $〈l,u〉$ which finishes the proof of part (iii). The statement (iv) follows immediately from the definitions of functions u and l. □

Of course, the set ${\mathcal{P}}^2\left(Y\right)$ is partially ordered by the inclusion relation. Thus, there is a naturally induced partial order ⪯ on ${\mathcal{P}}^2{\left(Y\right)}^X$ in which one function dominates another if this is true pointwise, i.e., for ${\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2\in {\mathcal{P}}^2{\left(Y\right)}^X$, ${\mathrm{\Psi }}_1⪯{\mathrm{\Psi }}_2$ means that ${\mathrm{\Psi }}_1\left(x\right)\subset {\mathrm{\Psi }}_2\left(x\right)$ for all $x\in X$.

Analogously, the partial order ⪯ on ${\mathcal{P}}^2{\left(Y\right)}^X$ induces a partial order on $\mathcal{C}.\mathcal{O}(X,Y)$, which we will also denote by ⪯, i.e.,

if $〈h_{2n},h_{2n-1},...,h_2,h_1〉$ and $〈g_{2m},g_{2m-1},...,g_2,g_1〉$ belong to $\mathcal{C}.\mathcal{O}(X,Y)$, then

$〈h_{2n},h_{2n-1},...,h_2,h_1〉⪯〈g_{2m},g_{2m-1},...,g_2,g_1〉$ just when

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉⪯〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉$

for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ i.e.,

for every $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $x\in X$, the following inclusion holds

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉\left(x\right)\subset 〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉\left(x\right)$.

Now, let us recall some definitions. By a monoid, we mean a semigroup with a neutral element. A partially ordered monoid [11] is an ordered quadruple $(\mathcal{S},\odot ,e,⪯)$ such that

(i) $(\mathcal{S},\odot ,e)$ is a monoid,

(ii) $(\mathcal{S},⪯)$ is a partially ordered set and

(iii) the order ⪯ is compatible with ⊙, in the sense that for all $a,b,c\in \mathcal{S}$, $a⪯b$ implies $a\odot c⪯b\odot c$ and $c\odot a⪯c\odot b$.

Lemma 8. 

For any topological spaces $(X,\tau )$ and $(Y,\sigma )$, the set $\mathcal{C}.\mathcal{O}(X,Y)$ has the structure of a partially ordered monoid $(\mathcal{C}.\mathcal{O}(X,Y),\odot ,〈...〉,⪯)$ under the binary operation ⊙ defined by

$〈h_{2n},h_{2n-1},...,h_2,h_1〉\odot 〈g_{2m},g_{2m-1},...,g_2,g_1〉\left(\mathrm{\Psi }\right)$=

$〈h_{2n},h_{2n-1},...,h_2,h_1,〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉〉$

for any $〈h_{2n},h_{2n-1},...,h_2,h_1〉$, $〈g_{2m},g_{2m-1},...,g_2,g_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$ and a function $\mathrm{\Psi }:X⟶{\mathcal{P}}^2\left(Y\right)$.

Proof. 

According to Lemma 7 (i), for any

$〈h_{2n},h_{2n-1},...,h_1〉,〈g_{2m},g_{2m-1},...,g_1〉$, $〈p_{2k},p_{2k-1},...,p_2〉\in \mathcal{C}.\mathcal{O}(X,Y)$ we have

$(〈h_{2n},h_{2n-1},...,h_2,h_1〉\odot 〈g_{2m},g_{2m-1},...,g_2,g_1〉)\odot 〈p_{2k},p_{2k-1},...,p_2,p_1〉=$

$〈h_{2n},h_{2n-1},...,h_2,h_1,g_{2m},g_{2m-1},...,g_2,g_1,p_{2k},p_{2k-1},...,p_2,p_1〉=$

$〈h_{2n},h_{2n-1},...,h_2,h_1〉\odot (〈g_{2m},g_{2m-1},...,g_2,g_1〉\odot 〈p_{2k},p_{2k-1},...,p_2,p_1〉)$. So, the operation ⊙ fulfills the associative law.

The function $〈...〉:{\mathcal{P}}^2{\left(Y\right)}^X⟶{\mathcal{P}}^2{\left(Y\right)}^X$ defined by $〈...〉\left(\mathrm{\Psi }\right)=〈\mathrm{\Psi }〉$ for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$, is the neutral element for the operation ⊙.

Indeed, applying Lemma 7 (i), we get

$〈h_{2n},h_{2n-1},...,h_2,h_1〉\odot 〈...〉\left(\mathrm{\Psi }\right)$ = $〈h_{2n},h_{2n-1},...,h_2,h_1,〈\mathrm{\Psi }〉〉=$

$〈h_{2n},h_{2n-1},...,h_2,h_1〉\left(\mathrm{\Psi }\right)$

for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $〈h_{2n},h_{2n-1},..,h_2,h_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$.

In the reverse order, we have

$〈...〉\odot 〈h_{2n},h_{2n-1},...,h_1〉\left(\mathrm{\Psi }\right)=$

$〈〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉〉=\overline{〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉}=$

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉=〈h_{2n},h_{2n-1},...,h_2,h_1〉\left(\mathrm{\Psi }\right)$ because, according to Lemma 2, the values of cluster functions are closed in the space ($\mathcal{P}\left(Y\right),{\sigma }^u)$.

We will now show that the relation ⪯ is compatible with ⊙. For this purpose, let us take two arbitrary cluster operators $〈h_{2n},h_{2n-1},...,h_2,h_1〉$ and $〈g_{2m},g_{2m-1},...,g_2,g_1〉$ such that $〈h_{2n},h_{2n-1},...,h_2,h_1〉⪯〈g_{2m},g_{2m-1},...,g_2,g_1〉$. So, according to the definition,

$〈h_{2n},h_{2n-1},...,h_2,h_1〉\left(\mathrm{\Psi }\right)⪯〈g_{2m},g_{2m-1},...,g_2,g_1〉\left(\mathrm{\Psi }\right)$ i.e.,

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉⪯〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉$$(*)$

for every $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$.

Of course, it holds for every function $\mathrm{\Psi }$ such that $\mathrm{\Psi }=〈s_{2k},s_{2k-1},...,s_2,s_1,{\mathrm{\Psi }}^{*}〉$, where ${\mathrm{\Psi }}^{*}\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $〈s_{2k},s_{2k-1},...,s_2,s_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$. Thus, for every $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $〈s_{2k},s_{2k-1},...,s_2,s_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$ we have

$〈h_{2n},h_{2n-1},...,h_2,h_1,〈s_{2k},s_{2k-1},...,s_2,s_1,\mathrm{\Psi }〉〉⪯$

$〈g_{2m},g_{2m-1},...,g_2,g_1,〈s_{2k},s_{2k-1},...,s_2,s_1,\mathrm{\Psi }〉〉$ which means that

$〈h_{2m},h_{2m-1},...,h_2,h_1,s_{2k},s_{2k-1},...,s_2,s_1〉\left(\mathrm{\Psi }\right)⪯$

$〈g_{2m},g_{2m-1},...,g_2,g_1,s_{2k},s_{2k-1},...,s_2,s_1〉\left(\mathrm{\Psi }\right)$ or equivalently,

$〈h_{2m},h_{2m-1},...,h_2,h_1〉\odot 〈s_{2k},s_{2k-1},...,s_2,s_1〉\left(\mathrm{\Psi }\right)⪯$

$〈g_{2m},g_{2m-1},...,g_2,g_1〉\odot 〈s_{2k},s_{2k-1},...,s_2,s_1〉\left(\mathrm{\Psi }\right)$. So,

$〈h_{2m},h_{2m-1},...,h_2,h_1〉\odot 〈s_{2k},s_{2k-1},...,s_2,s_1〉⪯$

$〈g_{2m},g_{2m-1},...,g_2,g_1〉\odot 〈s_{2k},s_{2k-1},...,s_2,s_1〉$. $(**)$

Now, let us note that the assumption $(*)$ means that

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉\left(x\right)\subset 〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉\left(x\right)$ for all $x\in X$. Consequently, according to Lemma 7 (iv), for every $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $x\in X$ we have

$〈s_{2k},s_{2k-1},...,s_2,s_1,〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉〉\left(x\right)\subset$

$〈s_{2k},s_{2k-1},...,s_2,s_1,〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉〉\left(x\right)$. So,

$〈s_{2k},s_{2k-1},...,s_2,s_1,〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉〉⪯$

$〈s_{2k},s_{2k-1},...,s_2,s_1,〈g_{2m},g_{2m-1},...,g_2,g_1,\mathrm{\Psi }〉〉$ for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$, i.e.,

$〈s_{2k},s_{2k-1},...,s_2,s_1〉\odot 〈h_{2n},h_{2n-1},...,h_1〉⪯$

$〈s_{2k},s_{2k-1},...,s_2,s_1〉\odot 〈g_{2m},g_{2m-1},...,g_2,g_1〉$ which, according to $(**)$, completes the proof of compatibility between ⪯ and ⊙. □

Theorem 1. 

For any topological spaces $(X,\tau )$ and $(Y,\sigma )$, the monoid $\mathcal{C}.\mathcal{O}(X,Y)$ has at most nine elements related to each other, as shown in the diagram below, where the arrows are compatible with the relation ⪯.

Proof. 

When it comes to the diagram, let us first prove that

$•〈\mathcal{I}〉⪯〈\mathcal{I},\mathcal{S},\mathcal{I}〉$,

$•〈\mathcal{I},\mathcal{S},\mathcal{I}〉⪯〈\mathcal{S},\mathcal{I}〉$ and $〈\mathcal{I},\mathcal{S},\mathcal{I}〉⪯〈\mathcal{I},\mathcal{S}〉$,

$•〈\mathcal{S},\mathcal{I}〉⪯〈\mathcal{S},\mathcal{I},\mathcal{S}〉$ and $〈\mathcal{I},\mathcal{S}〉⪯〈\mathcal{S},\mathcal{I},\mathcal{S}〉$ and,

$•〈\mathcal{S},\mathcal{I},\mathcal{S}〉⪯〈\mathcal{S}〉$ i.e.,

for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $x\in X$, the following hold:

(a) $〈\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\cap 〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ and

(b) $〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\cup 〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)$.

For that purpose, let us note that

-$〈\mathcal{I},\mathrm{\Psi }〉\left(x\right)\in 〈\mathcal{I},\mathrm{\Psi }〉\left(U\right)\in 〈\mathcal{I},\mathrm{\Psi }〉\left(\tau \left(x\right)\right)$ and

-$〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)\in 〈\mathcal{S},\mathrm{\Psi }〉\left(U\right)\in 〈\mathcal{S},\mathrm{\Psi }〉\left(\tau \left(x\right)\right)$

for any $x\in X$ and $U\in \tau \left(x\right)$.

Therefore, using Lemma 3, we obtain

$〈\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)$ and $〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ for all $x\in X$, respectively. Hence, since according to Lemma 7 (iii) and (iv), we have $〈\mathcal{I},\mathcal{I}〉$ = $〈\mathcal{I}〉$ and $〈\mathcal{S},\mathcal{S}〉$ = $〈\mathcal{S}〉$ and therefore $〈\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)$ and $〈\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ for all $x\in X$ i.e., the first inclusion in (a) and the second inclusion in (b) are true.

Now let us observe that, by Lemma 3, we have $〈\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset \overline{\mathrm{\Psi }\left(x\right)}\subset 〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ or equivalently, $〈\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ for all $x\in X$ since $\mathrm{\Psi }\left(x\right)\in \mathrm{\Psi }\left(U\right)\in \mathrm{\Psi }\left(\tau \right(x\left)\right)$ for all $x\in X$ and $U\in \tau \left(x\right)$. So, according to Lemma 7 (iv) and (i), we have

-$〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ and $〈\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$

for all $x\in X$.

Consequently, again from Lemma 7 (iv), we obtain

-$〈\mathcal{I},\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ and $〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$

for all $x\in X$.

We will finish the proof of (a) and (b) by showing, in an entirely analogous way, that

-$〈\mathcal{I},\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)$ and $〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ for all $x\in X$.

Indeed, since it is clear that

-$〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\in 〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(U\right)\in 〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(\tau \left(x\right)\right)$ and

-$〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\in 〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(U\right)\in 〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(\tau \left(x\right)\right)$

for all $x\in X$ and $U\in \tau \left(x\right)$,

applying Lemma 2.3 we obtain

-$〈\mathcal{I},\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathrm{\Psi }〉\left(x\right)$ and

-$〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ respectively,

for all $x\in X$.

We now show that the set $\left\{〈\mathcal{I}〉,〈\mathcal{S},〉,〈\mathcal{S},\mathcal{I}〉,〈\mathcal{I},\mathcal{S}〉,〈\mathcal{S},\mathcal{I},\mathcal{S}〉,〈\mathcal{I},\mathcal{S},\mathcal{I}〉\right\}$ is closed under the operation ⊙ as presents the following table, where the factor that labels the row comes first.

⊙	$〈\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S}〉$
$〈\mathcal{I}〉$	$〈\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S}〉$	$〈\mathcal{I},\mathcal{S}〉$	$〈\mathcal{I},\mathcal{S}〉$
$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I},〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S}〉$	$〈\mathcal{I},\mathcal{S}〉$	$〈\mathcal{I},\mathcal{S}〉$
$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$
$〈\mathcal{I},\mathcal{S}〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S},\mathcal{I}〉$	$〈\mathcal{I},\mathcal{S}〉$	$〈\mathcal{I},\mathcal{S}〉$	$〈\mathcal{I},\mathcal{S}〉$
$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$
$〈\mathcal{S}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S},\mathcal{I},\mathcal{S}〉$	$〈\mathcal{S}〉$

It is enough to verify the following two pairs of equalities:

(c) $〈h_{2n},...,h_1,\mathcal{I},\mathcal{I},g_{2m},...,g_1〉$= $〈h_{2n},...,h_1,\mathcal{I},g_{2m},...,g_1〉$ and

$〈h_{2n},...,h_1,\mathcal{S},\mathcal{S},g_{2m},...,g_1〉$= $〈h_{2n},...,h_1,\mathcal{S},g_{2m},...,g_1〉$

for any $〈h_{2n},h_{2n-1},...,h_1〉$, $〈g_{2m},g_{2m-1},...,g_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$ and,

(d) $〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S}〉=〈\mathcal{I},\mathcal{S}〉$ and $〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}〉=〈\mathcal{S},\mathcal{I}〉$.

For the proof of (c), let us note that by applying Lemma 7 (iii), we have $〈\mathcal{S},\mathcal{S}〉$ = $〈\mathcal{S}〉$ and $〈\mathcal{I},\mathcal{I}〉$ = $〈\mathcal{I}〉$ and, according to part (i) of this lemma we get

$〈h_{2n},h_{2n-1},...,h_1,\mathcal{I},\mathcal{I},g_{2m},g_{2m-1},...,g_1〉\left(\mathrm{\Psi }\right)=$

$〈h_{2n},h_{2n-1},...,h_1,〈\mathcal{I},\mathcal{I},g_{2m},g_{2m-1},...,g_1,\mathrm{\Psi }〉〉=$

$〈h_{2n},h_{2n-1},...,h_1,〈\mathcal{I},\mathcal{I},〈g_{2m},g_{2m-1},...,g_1,\mathrm{\Psi }〉〉〉=$

$〈h_{2n},h_{2n-1},...,h_1,〈\mathcal{I},〈g_{2m},g_{2m-1},...,g_1,\mathrm{\Psi }〉〉〉=$

$〈h_{2n},h_{2n-1},...,h_1,\mathcal{I},〈g_{2m},g_{2m-1},...,g_1,\mathrm{\Psi }〉〉=$

$〈h_{2n},h_{2n-1},...,h_1,\mathcal{I},g_{2m},g_{2m-1},...,g_1〉\left(\mathrm{\Psi }\right)$ for any $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$.

In the same way, one can check the second part of (c).

Now, since condition (b) implies that $〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)$, by Lemma 7 (iv) we have $〈\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{S},\mathrm{\Psi }〉\left(x\right)=〈\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ and consequently,

-$〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $x\in X$.

On the other hand, for the same reason as the above, we obtain

$〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ and consequently,

-$〈\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S},\mathrm{\Psi }〉\left(x\right)$ for all $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $x\in X$.

This ends the proof of the first equality in (d).

The checking of the second equality proceeds analogously by using condition (a) instead of (b).

To finish the proof, it suffices to note that for all $〈h_{2n},h_{2n-1},..,h_1〉\in \mathcal{C}.\mathcal{O}(X,Y)$, $\mathrm{\Psi }\in {\mathcal{P}}^2{\left(Y\right)}^X$ and $x\in X$, the following two properties hold:

• $〈u,u,\mathrm{\Psi }〉\left(x\right)\subset 〈h_{2n},h_{2n-1},...,h_1,\mathrm{\Psi }〉\left(x\right)\subset 〈l,l,\mathrm{\Psi }〉\left(x\right)$ and

• if $(h_{2i},h_{2i-1})\in \left\{(u,u),(l,l)\right\}$ for some $i\in \left\{1,2,...n\right\}$, then

$〈h_{2n},h_{2n-1},...,h_1,\mathrm{\Psi }〉\left(x\right)=〈u,u,\mathrm{\Psi }〉\left(x\right)$ or

$〈h_{2n},h_{2n-1},...,h_1,\mathrm{\Psi }〉\left(x\right)=〈l,l,\mathrm{\Psi }〉\left(x\right)$.

The first property is an obvious consequence of Remark 4 and Lemma 2. When it comes to the second property, according to Remark 4, applying Lemma 7 (i) and then (ii) parts (a) and (c), we obtain

$〈h_{2n},h_{2n-1},...,h_{2i+1},u,u,h_{2i-2},...,h_1,\mathrm{\Psi }〉=$

$〈h_{2n},h_{2n-1},...,h_{2i+1},〈u,u,h_{2i-2},...,h_1,\mathrm{\Psi }〉〉=$

$〈h_{2n},h_{2n-1},...,h_{2i+1},〈u,u,\mathrm{\Psi }〉〉=〈u,u,\mathrm{\Psi }〉$.

The proof for the case $(h_{2i},h_{2i-1})=(l,l)$ follows exactly in the same manner. So, the proof of Theorem 1 is finished. □

2.3. Generalized Continuity in Terms of the Cluster Operators

The concept of cluster functions of the form $〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉$ is closely related to the classical notion of cluster sets of a single-valued function $f:(X,\tau )\to (Y,\sigma )$ at a point $x\in X$ [9], [12] defined by

$C_f\left(x\right)=\bigcap \left\{Cl\left(f\right(U\left)\right)\subset Y:U\in \tau \left(x\right)\right\}$.

In [8], this concept has naturally been extended to multifunctions $F:(X,\tau )\to (Y,\sigma )$ as

$C_F\left(x\right)=\bigcap \left\{Cl\left(F\right(U\left)\right)\subset Y:U\in \tau \left(x\right)\right\}$.

It is easy to check that

$C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Cl(F^{-}\left(W\right)\right\}$.

According to Remark 2, we use the functions defined as

$x⟶\overline{\left\{F\left(x\right)\right\}}$ and $x⟶\overline{\mathcal{P}\left(F\right(x\left)\right)}$ for all $x\in X$.

Using Lemma 5, we can see that these functions may be presented in the form of patterns through the operations $\mathcal{I}$ and $\mathcal{S}$ as

$\overline{\left\{F\right(x\left)\right\}}=\mathcal{I}\left(\left[{\mathcal{P}}^2\left(F\left(x\right)\right)\right]\right)$ and $\overline{\mathcal{P}\left(F\right(x\left)\right)}=\mathcal{S}\left(\left[{\mathcal{P}}^2\left(F\left(x\right)\right)\right]\right)$ for all $x\in X$.

We will use the shorter notation, namely $\mathcal{I}F$ and $\mathcal{S}F$, respectively i.e.,

$\mathcal{I}F\left(x\right)=\overline{\left\{F\right(x\left)\right\}}$ and $\mathcal{S}F\left(x\right)=\overline{\mathcal{P}\left(F\right(x\left)\right)}$ for all $x\in X$.

Of course, $\mathcal{I}F,\mathcal{S}F\in {\mathcal{P}}^2{\left(Y\right)}^X$ so, one can consider the cluster functions of types $〈h_{2n},h_{2n-1},...,h_2,h_1,\mathcal{I}F〉$ and $〈h_{2n},h_{2n-1},...,h_2,h_1,\mathcal{S}F〉$.

Using the concept of cluster functions, we can characterize cluster sets as follows

Remark 8. 

For any multifunction $F:(X,\tau )\to (Y,\sigma )$ and $x\in X$, the following properties are equivalent:

(i) $y\in C_F\left(x\right)$ and

(ii) $\left\{y\right\}\in 〈\mathcal{S},\mathcal{S}F〉\left(x\right)$.

Indeed, it is enough to use, according to the characterization of $\mathcal{S}$ given in Lemma 2, the equality $〈\mathcal{S},\mathcal{S}F〉\left(x\right)={\bigcap }_{U_1\in \tau \left(x\right)}\overline{{\bigcup }_{x_1\in U_1}\mathcal{P}\left(F\left(x_1\right)\right)}$. Next, according to Remark 1, the property (ii) means that for every open sets W and $U_1$ with $\left\{y\right\}\in \mathcal{P}\left(W\right)$ and $x\in U_1$, there exists $x_1\in U_1$ such that $\mathcal{P}\left(W\right)\cap \mathcal{P}\left(F\left(x_1\right)\right)\ne \varnothing$ i.e., $x_1\in F^{-}\left(W\right)$.

The analogous characterizations we have in the case of the other types of cluster sets listed below that are investigated in [13] as an extension of the concept studied in [14] and [15]. Those types of cluster sets describe many kinds of generalized continuity of multifunctions [13], and for many types of convergence of nets of multifunctions, there are strict relations between the cluster set of the limit multifunction and the appropriate cluster sets of the members of the nets of multifunctions [16].

• $u.\alpha C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Int\left(Cl\left(Int\left(F^{+}\left(W\right)\right)\right)\right)\right\}$,

• $l.\alpha C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Int\left(Cl\left(Int\left(F^{-}\left(W\right)\right)\right)\right)\right\}$,

• $u.q.C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Cl\left(Int\left(F^{+}\left(W\right)\right)\right)\right\}$,

• $l.q.C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Cl\left(Int\left(F^{-}\left(W\right)\right)\right)\right\}$,

• $u.p.C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Int\left(Cl\left(F^{+}\left(W\right)\right)\right)\right\}$,

• $l.p.C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Int\left(Cl\left(F^{-}\left(W\right)\right)\right)\right\}$,

• $u.\beta C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Cl\left(Int\left(Cl\left(F^{+}\left(W\right)\right)\right)\right)\right\}$,

• $l.\beta C_F\left(x\right)=\left\{y\in Y:x\in {\bigcap }_{W\in \sigma \left(y\right)}Cl\left(Int\left(Cl\left(F^{-}\left(W\right)\right)\right)\right)\right\}$.

In the case of single-valued functions $f:(X,\tau )⟶(Y,\sigma )$ that are interpreted as multifunctions F defined by $F\left(x\right)=\left\{f\left(x\right)\right\}$ for all $x\in X$, we have the following suitable definitions of cluster sets:

• $\alpha .C_f\left(x\right)=l.\alpha C_F\left(x\right)=l.\alpha C_F\left(x\right)$.

• $q.C_f\left(x\right)=l.q.C_F\left(x\right)=u.q.C_F\left(x\right)$,

• $p.C_f\left(x\right)=l.p.C_F\left(x\right)=u.p.C_F\left(x\right)$ and

• $\beta .C_f\left(x\right)=l.\beta C_F\left(x\right)=u.\beta C_F\left(x\right)$.

The concept of cluster sets of type $q.C_f\left(x\right)$ was introduced in [14] and used later in [15] and [17]. Another type of cluster set [18], of multifunctions $F:(X,\tau )⟶(Y,\sigma )$ has been defined as follows:

If $\mathcal{B}$ is a non-empty family of non-empty subsets of X, then a point $y\in Y$ is called a $\mathcal{B}$-cluster point of F at $x\in X$, i.e., $y\in {\mathcal{B}}_F\left(x\right)$, if for any open sets U, V with $x\in U$ and $y\in V$, there exists $B\in \mathcal{B}$ such that $B\subset U$ and $B\subset F^{-}\left(W\right)$.

This concept is used in further investigations e.g., [18,19,20,21,22,23,24,25] and, it describes two of those listed above types. Namely, it is easy to show that in the case when $\mathcal{B}$ is the family of all nonempty open subsets $B\subset X$, then ${\mathcal{B}}_F\left(x\right)=l.q.C_F\left(x\right)$ and, the equality is also true if $\mathcal{B}$ is the family of all nonempty $\alpha$-open [26] (resp. semi-open [27]) subsets $A\subset X$ defined by the condition $A\subset Int\left(Cl\right(Int\left(A\right)\left)\right)$ (resp. $A\subset Cl\left(Int\right(A)$). Analogously, in the case when $\mathcal{B}$ is the family of all nonempty pre-open (locally dense) subsets $A\subset X$ [28], ([29]) defined by the condition $A\subset Int\left(Cl\right(A\left)\right))$, we have ${\mathcal{B}}_F\left(x\right)=l.\beta .C_F\left(x\right)$ and, the equality is also true if $\mathcal{B}$ is the family of all nonempty $\beta$-open [30] subsets $A\subset X$ defined by the condition $A\subset Cl\left(Int\right(Cl\left(A\right)\left)\right)$.

Cluster sets may be also considered as the values of multifunctions from $(X,\tau )$, to $(Y,\sigma )$. Such multifunctions have been used in [19,20,31].

Analogously to Remark 2.15, the following simple observation shows the connection between the concepts of cluster functions of the forms

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathcal{I}F〉$ and $〈h_{2n},h_{2n-1},...,h_2,h_1,\mathcal{S}F〉$, and cluster sets.

Remark 9. 

For any multifunction $F:(X,\tau )\to (Y,\sigma )$, $x\in X$ and $y\in Y$, the following equivalences hold:

$\left(i\right)$$y\in l.\alpha .C_F\left(x\right)$ if and only if $\left\{y\right\}\in 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)$

(resp. $y\in u.\alpha .C_F\left(x\right))$ if and only if and $\left\{y\right\}\in 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right))$,

$\left(ii\right)$$y\in l.q.C_F\left(x\right)$ if and only if $\left\{y\right\}\in 〈\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)$

(resp. $y\in u.q.C_F\left(x\right))$ if and only if and $\left\{y\right\}\in 〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right))$,

$\left(iii\right)$$y\in l.p.C_F\left(x\right)$ if and only if $\left\{y\right\}\in 〈\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)$

(resp. $y\in u.p.C_F\left(x\right))$ if and only if and $\left\{y\right\}\in 〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right))$,

$\left(iv\right)$$y\in l.\beta .C_F\left(x\right)$ if and only if $\left\{y\right\}\in 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)$

(resp. $y\in u.\beta .C_F\left(x\right))$ if and only if and $\left\{y\right\}\in 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right))$.

Proof. 

Let us note that, according to the characterizations of $\mathcal{I}$ and $\mathcal{S}$ given in Lemma 2, we have the following equalities:

(${\alpha }_l$) $〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)$=

$\overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}{\bigcap }_{U_2\in \tau \left(x_1\right)}\overline{{\bigcup }_{x_2\in U_2}{\bigcup }_{U_3\in \tau \left(x_2\right)}{\bigcap }_{x_3\in U_3}\overline{\mathcal{P}\left(F\left(x_3\right)\right)}}}$,

(${\alpha }_u$) $〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$=

$\overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}{\bigcap }_{U_2\in \tau \left(x_1\right)}\overline{{\bigcup }_{x_2\in U_2}{\bigcup }_{U_3\in \tau \left(x_2\right)}{\bigcap }_{x_3\in U_3}\overline{\left\{F\left(x_3\right)\right\}}}}$,

($q_l$) $〈\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)$= ${\bigcap }_{U_1\in \tau \left(x\right)}\overline{{\bigcup }_{x_1\in U_1}{\bigcup }_{U_2\in \tau \left(x_1\right)}{\bigcap }_{x_2\in U_2}\overline{\mathcal{P}\left(F\left(x_2\right)\right)}}$,

($q_u$) $〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$= ${\bigcap }_{U_1\in \tau \left(x\right)}\overline{{\bigcup }_{x_1\in U_1}{\bigcup }_{U_2\in \tau \left(x_1\right)}{\bigcap }_{x_2\in U_2}\overline{\left\{F\left(x_2\right)\right\}}}$,

($p_l$) $〈\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)$=$\overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}{\bigcap }_{U_2\in \tau \left(x_1\right)}\overline{{\bigcup }_{x_2\in U_2}\mathcal{P}\left(F\left(x_2\right)\right)}}$,

($p_u$) $〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$= $\overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}{\bigcap }_{U_2\in \tau \left(x_1\right)}\overline{{\bigcup }_{x_2\in U_2}\left\{F\left(x_2\right)\right\}}}$,

(${\beta }_l$) $〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)$=

${\bigcap }_{U_1\in \tau \left(x\right)}\overline{{\bigcup }_{x_1\in U_1}{\bigcup }_{U_2\in \tau \left(x_1\right)}{\bigcap }_{x_2\in U_2}{\bigcap }_{U_3\in \tau \left(x_2\right)}\overline{{\bigcup }_{x_3\in U_3}\mathcal{P}\left(F\left(x_3\right)\right)}}$,

(${\beta }_l$) $〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$=

${\bigcap }_{U_1\in \tau \left(x\right)}\overline{{\bigcup }_{x_1\in U_1}{\bigcup }_{U_2\in \tau \left(x_1\right)}{\bigcap }_{x_2\in U_2}{\bigcap }_{U_3\in \tau \left(x_2\right)}\overline{{\bigcup }_{x_3\in U_3}\left\{F\left(x_3\right)\right\}}}$.

Now, it is enough to use Remark 1 and then, we immediately obtain the following equivalences for each $W\in \sigma$:

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)\ne \varnothing$ if and only if $x\in Int(Cl\left(Int\left(F^{-}\left(W\right)\right)\right)$,

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{I},S,\mathcal{I},\mathcal{I}F〉\left(x\right)\ne \varnothing$ if and only if $x\in Int(Cl\left(Int\left(F^{+}\left(W\right)\right)\right)$,

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)\ne \varnothing$ if and only if $x\in Cl(Int\left(F^{-}\left(W\right)\right)$,

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)\ne \varnothing$ if and only if $x\in Cl(Int\left(F^{+}\left(W\right)\right)$,

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)\ne \varnothing$ if and only if $x\in Int(Cl\left(F^{-}\left(W\right)\right)$,

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)\ne \varnothing$ if and only if $x\in Int(Cl\left(F^{+}\left(W\right)\right)$,

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)\ne \varnothing$ if and only if $x\in Cl(Int\left(Cl\left(F^{-}\left(W\right)\right)\right)$,

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)\ne \varnothing$ if and only if $x\in Cl(Int\left(Cl\left(F^{+}\left(W\right)\right)\right)$.

□

The cluster functions of the form $〈h_{2n},h_{2n-1},...,h_2,h_1,\mathrm{\Psi }〉:X\to {\mathcal{P}}^2\left(Y\right)$ are convenient tools to characterize the properties related to the continuity of multifunctions. By way of illustration, let us recall the classical types of continuity for multifunctions.

The continuity of a multifunction is defined by its upper and lower continuity [32]. A multifunction $F:(X,\tau )\to (Y,\sigma )$ is said to by upper semi continuous (briefly $u.s.c.$) (resp. lower semi continuous (briefly $l.s.c.$)) at a point $x\in X$ if whenever W is an open subset of Y such that $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$, then $x\in Int\left(F^{+}\left(W\right)\right)$ (resp. $x\in Int\left(F^{-}\left(W\right)\right))$. The set of all such points will be denoted by $C_u\left(F\right)$ (resp. $C_l\left(F\right)$). A multifunction F is called $u.s.c.$ (resp. $l.s.c.$) if $C_u\left(F\right)=X$ (resp. $C_l\left(F\right)=X$).

It is easy to see that a multifunction $F:(X,\tau )\to (Y,\sigma )$ is $u.s.c.$ (resp. $l.s.c.$) if and only if it is continuous when it is considered to be a single-valued function $F:(X,\tau )\to (\mathcal{P}\left(Y\right),{\sigma }^u)$ (resp. $F:(X,\tau )\to (\mathcal{P}\left(Y\right),{\sigma }^l)$) [3,33].

By using the concept of cluster functions of the forms

$〈h_{2n},h_{2n-1},...,h_2,h_1,\mathcal{I}F〉$ and $〈h_{2n},h_{2n-1},...,h_2,h_1,\mathcal{S}F〉$, both these types of continuity can be characterized in terms of the upper Vietoris topology as follows:

Lemma 9. 

A multifunction $F:(X,\tau )\to (Y,\sigma )$ is upper semi continuous (resp. lower semi continuous) at a point $x\in X$ if and only if

$〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{I}F〉\left(x\right)$(resp. $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S}F〉\left(x\right))$.

Proof. 

Since, according to Lemma 2, we have

($c_u$) $〈\mathcal{I},\mathcal{I}F〉\left(x\right)$=$\overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}\overline{\left\{F\left(x_1\right)\right\}}}$ and

($c_l$) $〈\mathcal{I},\mathcal{S}F〉\left(x\right)$=$\overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}\overline{\mathcal{P}\left(F\left(x_1\right)\right)}}$.

So, the following pairs of statements are equivalent for each $W\in \sigma$ and $x\in X$:

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{I},\mathcal{I}F〉\left(x\right)\ne \varnothing$ and $x\in Int\left(F^{+}\left(W\right)\right)$,

• $\mathcal{P}\left(W\right)\cap 〈\mathcal{I},\mathcal{S}F〉\left(x\right)\ne \varnothing$ and $x\in Int\left(F^{-}\left(W\right)\right)$.

Of course, $\mathcal{P}\left(W\right)\cap 〈\mathcal{I}F〉\left(x\right)\ne \varnothing$ (resp. $\mathcal{P}\left(W\right)\cap 〈\mathcal{S}F〉\left(x\right)\ne \varnothing )$ means that $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right)$). Hence, the characterization is true. □

Remark 10. 

One can also consider the other two possible relations, namely

$〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S}F〉\left(x\right)$ and

$〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{I}F〉\left(x\right)$ for $x\in X$.

We denote those types of continuity as $u.l.s.c.$ and $l.u.s.c.$, respectively. The set of all such points will be denoted by $C_{ul}\left(F\right)$ and $C_{lu}\left(F\right)$), respectively. Of course, the first of these properties is equivalent to $x\in Int\left(F^{-}\left(W\right)\right)$ for any open subset W such that $x\in F^{+}\left(W\right)$, which defines the type of continuity introduced in [34].

The second property means that $x\in Int\left(F^{+}\left(W\right)\right)$ for any open subset W such that $x\in F^{-}\left(W\right)$ or equivalently, F is $u.s.c.$ at x and $F\left(x\right)$ is a singleton.

Analogous characterizations one can formulate for many types of generalized continuity. Let us first quote the definitions of some classical types of them.

Definition 2. 

A multifunction $F:(X,\tau )\to (Y,\sigma )$ is called

• $u.\alpha .c.$ (resp. $l.\alpha .c.$)[35] at a point $x\in X$ i.e.,

$x\in \alpha .C_u\left(F\right)$ (resp. $x\in \alpha .C_l\left(F\right)$), if $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$

implies $x\in Int\left(Cl\left(Int\left(F^{+}\left(W\right)\right)\right)\right)$ (resp. $x\in Int\left(Cl\left(Int\left(F^{-}\left(W\right)\right)\right)\right))$

for all $W\in \sigma$,

• $u.q.c.$ (resp. $l.q.c.$)[36] at a point $x\in X$ i.e.,

$x\in q.C_u\left(F\right)$ (resp. $x\in q.C_l\left(F\right)$), if $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$

implies $x\in Cl\left(Int\left(F^{+}\left(W\right)\right)\right)$ (resp. $x\in Cl\left(Int\left(F^{-}\left(W\right)\right)\right))$

for all $W\in \sigma$,

• $u.p.c.$ (resp. $l.p.c.$)[37] at a point $x\in X$ i.e.,

$x\in p.C_u\left(F\right)$ (resp. $x\in p.C_l\left(F\right)$), if $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$

implies $x\in Int\left(Cl\left(F^{+}\left(W\right)\right)\right)$ (resp. $x\in Int\left(Cl\left(F^{-}\left(W\right)\right)\right))$

for all $W\in \sigma$,

• $u.\beta .c.$ (resp. $l.\beta .c.$)[38]) at a point x ∈ X i.e.,

$x\in \beta .C_u\left(F\right)$ (resp. $x\in \beta .C_l\left(F\right)$), if $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$

implies $x\in Cl\left(Int\left(Cl\left(F^{+}\left(W\right)\right)\right)\right)$ (resp. $x\in Cl\left(Int\left(Cl\left(F^{-}\left(W\right)\right)\right)\right))$

for all $W\in \sigma$.

In [39] the property $l.\beta .c.$ is called the lower demicontinuity ($l.d.c.$).

Analogously to the case of $u.s.c$ and $l.s.c.$, a multifunction F is $u.\alpha .c.$

(or. $l.\alpha .c.$) (resp. $u.q.c.$ (or $l.q.c.$), $u.p.c.$ (or $l.p.c.$), $u.\beta .c.$ (or $l.\beta .c.$)) if and only if it is $\alpha$-continuous (resp. semi-continuous, pre-continuous,

$\beta$-continuous) when it is considered to be a function $F:(X,\tau )\to (\mathcal{P}\left(Y\right),{\sigma }^u)$ (or $F:(X,\tau )\to (\mathcal{P}\left(Y\right),{\sigma }^l$)).

The requirements stated in the above forms of generalized continuity apply to upper inverse image $F^{+}\left(W\right)$ - in the type u (resp. lower inverse image $F^{-}\left(W\right)$ - in the type l) of open subsets W of Y. The use of mixed types of the inverse images leads to the following kinds of continuity.

Definition 3. 

A multifunction $F:(X,\tau )\to (Y,\sigma )$ is said to be

• $u.l.\alpha .c.$ (resp. $l.u.\alpha .c.$)[40] at a point $x\in X$ i.e.,

$x\in \alpha .C_{ul}\left(F\right)$ (resp. $x\in \alpha .C_{lu}\left(F\right)$), if $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$

implies $x\in Int\left(Cl\left(Int\left(F^{-}\left(W\right)\right)\right)\right)$ (resp. $x\in Int\left(Cl\left(Int\left(F^{+}\left(W\right)\right)\right)\right))$

for all $W\in \sigma$,

• $u.l.q.c.$ [40](or $l.u.q.c.$, [39]), at a point $x\in X$ i.e.,

$x\in q.C_{ul}\left(F\right)$ (resp. $x\in q.C_{lu}\left(F\right)$), if $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$

implies $x\in Cl\left(Int\left(F^{-}\left(W\right)\right)\right)$ (resp. $x\in Cl\left(Int\left(F^{+}\left(W\right)\right)\right))$

for all $W\in \sigma$,

• $u.l.p.c.$ (or $l.u.p.c.$, )[40], at a point $x\in X$ i.e.,

$x\in p.C_{ul}\left(F\right)$ (resp. $x\in p.C_{lu}\left(F\right)$), if $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$

implies $x\in Int\left(Cl\left(F^{-}\left(W\right)\right)\right)$ (resp. $x\in Int\left(Cl\left(F^{+}\left(W\right)\right)\right))$

for all $W\in \sigma$,

• $u.l.\beta .c.$ (resp. $l.u.\beta .c.$)[40] at a point $x\in X$ i.e.,

$x\in \beta .C_{ul}\left(F\right)$ (resp. $x\in \beta .C_{lu}\left(F\right)$), if $x\in F^{+}\left(W\right)$ (resp. $x\in F^{-}\left(W\right))$

implies $x\in Cl\left(Int\left(Cl\left(F^{-}\left(W\right)\right)\right)\right)$ (resp. $x\in Cl\left(Int\left(Cl\left(F^{+}\left(W\right)\right)\right)\right))$

for all $W\in \sigma$.

In [41] and [42], the $l.u.q.c.$ property was used under the name of minimality of $u.s.c.o.($u.s.c.with compact values) multifunction. But in [43,44] and [39],this property has been used independently of the condition $u.s.c.o.$

Of course, if a single-valued function $f:(X,\tau )\to (Y,\sigma )$ is treated as a multifunction F given by $F\left(x\right)=\left\{f\left(x\right)\right\}$ for all $x\in X$, then we have the following equalities:

• $C_u\left(F\right)=C_l\left(F\right)=C_{ul}\left(F\right)=C_{lu}\left(F\right)=C\left(f\right)$,

• $\alpha .C_u\left(F\right)=\alpha .C_l\left(F\right)=\alpha .C_{ul}\left(F\right)=\alpha .C_{lu}\left(F\right)=\alpha .C\left(f\right)$,

• $q.C_u\left(F\right)=q.C_l\left(F\right)=q.C_{ul}\left(F\right)=q.C_{lu}\left(F\right)=q.C\left(f\right)$,

• $p.C_u\left(F\right)=p.C_l\left(F\right)=p.C_{ul}\left(F\right)=p.C_{lu}\left(F\right)=p.C\left(f\right)$ and

• $\beta .C_u\left(F\right)=\beta .C_l\left(F\right)=\beta .C_{ul}\left(F\right)=\beta .C_{lu}\left(F\right)=\beta .C\left(f\right)$,

where $C\left(f\right)$ (resp. $\alpha .C\left(f\right)$, $q.C\left(f\right)$, $p.C\left(f\right)$, $\beta .C\left(f\right)$) denotes the set of all continuity (resp. $\alpha$-continuity [45], semi-continuity [46,27], pre-continuity [28], $\beta$-continuity [30]) points of f.

The following characterizations are analogous to those in Lemma 2.17.

Lemma 10. 

For any multifunction $F:(X,\tau )\to (Y,\sigma )$ and $x\in X$, the following equivalences hold:

$\left(\alpha \right)$• $x\in \alpha .C_u\left(F\right)$ if and only if $〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$,

• $x\in \alpha .C_l\left(F\right)$ if and only if $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)$,

• $x\in \alpha .C_{ul}\left(F\right)$ if and only if $〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)$,

• $x\in \alpha .C_{lu}\left(F\right)$ if and only if $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$,

$\left(q\right)$• $x\in q.C_u\left(F\right)$ if and only if $〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$,

• $x\in q.C_l\left(F\right)$ if and only if $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)$,

• $x\in q.C_{ul}\left(F\right)$ if and only if $〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right)$,

• $x\in q.C_{lu}\left(F\right)$ if and only if $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$,

$\left(p\right)$• $x\in p.C_u\left(F\right)$ if and only if $〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$,

• $x\in p.C_l\left(F\right)$ if and only if $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)$,

• $x\in p.C_{ul}\left(F\right)$ if and only if $〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)$,

• $x\in p.C_{lu}\left(F\right)$ if and only if $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$,

$\left(\beta \right)$• $x\in \beta .C_u\left(F\right)$ if and only if $〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$,

• $x\in \beta .C_l\left(F\right)$ if and only if $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)$,

• $x\in \beta .C_{ul}\left(F\right)$ if and only if $〈\mathcal{I}F〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right)$,

• $x\in \beta .C_{lu}\left(F\right)$ if and only if $〈\mathcal{S}F〉\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$.

Proof. 

It is enough to use the equivalences resulting from the characterizations (${\alpha }_l$), (${\alpha }_u$), ($q_l$), ($q_u$), ($p_l$), ($p_u$), (${\beta }_l$) and (${\beta }_u$) listed in Remark 9, and the equivalences listed in its proof. □

The following quite different types of generalized continuity can also be characterized using cluster operators.

Definition 4. 

A multifunction $F:(X,\tau )\to (Y,\sigma )$ is said to be

• $\alpha -$continuous [47,48] at a point $x\in X$ i.e., $x\in A.C\left(F\right)$, if

$x\in Int\left(Cl\left(Int(F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right))\right)\right)$ holds for all $W_1,W_2\in \sigma$

such that $x\in F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right)$,

• quasicontinuous [49] at a point $x\in X$ i.e., $x\in Q.C\left(F\right)$, if

$x\in Cl\left(Int(F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right))\right)$ holds for all $W_1,W_2\in \sigma$

such that $x\in F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right)$,

• precontinuous [47,48] at a point $x\in X$ i.e., $x\in P.C\left(F\right)$, if

$x\in Int\left(Cl(F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right))\right)$ holds for all $W_1,W_2\in \sigma$

such that $x\in F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right)$,

• $\beta -$continuous [48] at a point $x\in X$ i.e., $x\in B.C\left(F\right)$, if

$x\in Cl\left(Int\left(Cl(F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right))\right)\right)$ holds for all $W_1,W_2\in \sigma$

such that $x\in F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right)$.

With any multifunction $F:(X,\tau )\to (Y,\sigma )$ we consider the function

$\mathcal{I}F$×$\mathcal{S}F:X\to \mathcal{P}\left(\mathcal{P}\right(Y)\times \mathcal{P}(Y\left)\right)$ defined by

$\mathcal{I}F$×$\mathcal{S}F\left(x\right)=\overline{\left\{F\left(x\right)\right\}}\times \overline{\mathcal{P}\left(F\right(x\left)\right)}$ for all $x\in X$,

where $\mathcal{P}\left(Y\right)\times \mathcal{P}\left(Y\right)$ is equipped with the product topology derived from the upper Vietoris topology ${\sigma }^u$ on $\mathcal{P}\left(Y\right)$. So, it is clear that

$(\mathcal{P}\left(W_1\right)\times \mathcal{P}\left(W_2\right))\cap \mathcal{I}F$×$\mathcal{S}F\left(x\right)\ne \varnothing$ if and only if $x\in F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right)$ for all $W_1,W_2\in \sigma$ and $x\in X$.

We will use the following functions:

(A) $〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)$=

$\overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}{\bigcap }_{U_2\in \tau \left(x_1\right)}\overline{{\bigcup }_{x_2\in U_2}{\bigcup }_{U_3\in \tau \left(x_2\right)}{\bigcap }_{x_3\in U_3}\overline{\left\{F\left(x_3\right)\right\}}\times \overline{\mathcal{P}\left(F\left(x_3\right)\right)}}}$,

(Q) $〈\mathcal{S},\mathcal{I},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)$= ${\bigcap }_{U_1\in \tau \left(x\right)}\overline{{\bigcup }_{x_1\in U_1}{\bigcup }_{U_2\in \tau \left(x_1\right)}{\bigcap }_{x_2\in U_2}\overline{\left\{F\left(x_2\right)\right\}}\times \overline{\mathcal{P}\left(F\left(x_2\right)\right)}}$,

(P) $〈\mathcal{I},\mathcal{S},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)$= $\overline{{\bigcup }_{U_1\in \tau \left(x\right)}{\bigcap }_{x_1\in U_1}{\bigcap }_{U_2\in \tau \left(x_1\right)}\overline{{\bigcup }_{x_2\in U_2}\left\{F\left(x_2\right)\right\}\times \mathcal{P}\left(F\left(x_2\right)\right)}}$,

(B) $〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)$=

${\bigcap }_{U_1\in \tau \left(x\right)}\overline{{\bigcup }_{x_1\in U_1}{\bigcup }_{U_2\in \tau \left(x_1\right)}{\bigcap }_{x_2\in U_2}{\bigcap }_{U_3\in \tau \left(x_2\right)}\overline{{\bigcup }_{x_3\in U_3}\left\{F\left(x_3\right)\right\}\times \mathcal{P}\left(F\left(x_3\right)\right)}}$.

Thus, the following pairs of statements are equivalent for all $W_1,W_2\in \sigma$ and $x\in X$:

• $(\mathcal{P}\left(W_1\right)\times \mathcal{P}\left(W_2\right))\cap 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)\ne \varnothing$ and

$x\in Int\left(Cl\left(Int(F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right))\right)\right)$,

• $(\mathcal{P}\left(W_1\right)\times \mathcal{P}\left(W_2\right))\cap 〈\mathcal{S},\mathcal{I},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)\ne \varnothing$ and

$x\in Cl\left(Int(F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right))\right)$,

• $(\mathcal{P}\left(W_1\right)\times \mathcal{P}\left(W_2\right))\cap 〈\mathcal{I},\mathcal{S},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)\ne \varnothing$ and

$x\in Int\left(Cl(F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right))\right)$,

• $(\mathcal{P}\left(W_1\right)\times \mathcal{P}\left(W_2\right))\cap 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)\ne \varnothing$ and

$x\in Cl\left(Int\left(Cl(F^{+}\left(W_1\right)\cap F^{-}\left(W_2\right))\right)\right)$.

Hence we have the following characterizations:

Lemma 11. 

For any multifunction $F:(X,\tau )\to (Y,\sigma )$ and $x\in X$, the following equivalences hold:

• $x\in A.C\left(F\right)$ if and only if $\mathcal{I}F\times \mathcal{S}F\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)$,

• $x\in Q.C\left(F\right)$ if and only if $\mathcal{I}F\times \mathcal{S}F\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)$,

• $x\in P.C\left(F\right)$ if and only if $\mathcal{I}F\times \mathcal{S}F\left(x\right)\subset 〈\mathcal{I},\mathcal{S},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)$,

• $x\in B.C\left(F\right)$ if and only if $\mathcal{I}F\times \mathcal{S}F\left(x\right)\subset 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F\times \mathcal{S}F〉\left(x\right)$.

At the end of this chapter, we will characterize some types of continuity, originating in some generalized continuity of functions with the values in metric spaces.

A function f from a topological space $(X,\tau )$ to a metric space $(Y,d)$ is said to be cliquish at a point $x\in X$ [50,51], if for any $\epsilon >0$ and any open subset $U\subset X$ containing x there exists an open nonempty subset $G\subset U$ such that $d(f\left(x_1\right),f\left(x_2\right))<ϵ$ for all $x_1,x_2\in G$. Or equivalently,

$x\in Cl(\bigcup \left\{Int\left(f^{-1}\left(V\right)\right):V\in {\mathcal{V}}_{ϵ}\right\})$ for any $ϵ>0$,

where ${\mathcal{V}}_{ϵ}=\left\{B(y,ϵ):y\in Y\right\}$ and $B(y,ϵ)$ denotes the ball with center y and radius $ϵ$.

The appropriate topological form of this condition called $T_1-$cliquishness [52] (equivalently ${\chi }^1-$cliquish [53], (Proposition 2.3 (iii)), applies to functions f from a topological space $(X,\tau )$ to a topological space $(Y,\sigma )$ and is given by the property

• $x\in Cl(\bigcup \left\{Int\left(f^{-1}\left(V\right)\right):V\in \mathcal{V}\right\})$

for any open covering $\mathcal{V}$ of $(Y,\sigma )$.

It is a simple generalization of the notion of $T_1-$continuity [53] defined by

$x\in \bigcup \left\{Int\left(f^{-1}\left(V\right)\right):V\in \mathcal{V}\right\}$

for any open covering $\mathcal{V}$ of $(Y,\sigma )$.

The other types of topological forms of cliquishness [54] called pre ${\chi }^1-$cliquishness (resp. ${\chi }^1-s-$cliquishness, pre ${\chi }^1-s-$cliquishness, ${\chi }^1-\alpha -$cliquishness, pre ${\chi }^1-\alpha -$cliquishness) at a point $x\in$X, are defined by the following conditions:

• $x\in Cl(\bigcup \left\{Int\left(Cl\left(f^{-1}\left(V\right)\right)\right):V\in \mathcal{V}\right\})$ (resp.

• $x\in \bigcup \left\{Cl\left(Int\left(f^{-1}\left(V\right)\right)\right):V\in \mathcal{V}\right\}$,

• $x\in \bigcup \left\{Cl\left(Int\left(Cl\left(f^{-1}\left(V\right)\right)\right)\right):V\in \mathcal{V}\right\}$,

• $x\in \bigcup \left\{Int\left(Cl\left(Int\left(f^{-1}\left(V\right)\right)\right)\right):V\in \mathcal{V}\right\}$,

• $x\in \bigcup \left\{Int\left(Cl\left(f^{-1}\left(V\right)\right)\right):V\in \mathcal{V}\right\}$)

for any open covering $\mathcal{V}$ of $(Y,\sigma )$.

The extensions of those definitions to multifunctions have the following forms.

Definition 5. 

A multifunction $F:(X,\tau )\to (Y,\sigma )$ is said to be

• u-t-α-cliquish (resp. l-t-α-cliquish)[13] at a point $x\in X$ i.e.,

$x\in \alpha K_u\left(F\right)$ (resp. $x\in \alpha K_l\left(F\right)$), if $x\in \bigcup \left\{Int\left(Cl\left(Int\left(F^{+}\left(V\right)\right)\right)\right):V\in \mathcal{V}\right\}$ (resp. $x\in \bigcup \left\{Int\left(Cl\left(Int\left(F^{-}\left(V\right)\right)\right)\right):V\in \mathcal{V}\right\})$ for any open cover $\mathcal{V}$ of Y,

• u-t-q-cliquish (resp. l-t-q-cliquish)[13] at a point $x\in X$ i.e.,

$x\in q{K}_u\left(F\right)$ (resp. $x\in q{K}_l\left(F\right)$), if $x\in \bigcup \left\{Cl\left(Int\left(F^{+}\left(V\right)\right)\right):V\in \mathcal{V}\right\}$

(resp. $x\in \bigcup \left\{Cl\left(Int\left(F^{-}\left(V\right)\right)\right):V\in \mathcal{V}\right\})$ for any open cover $\mathcal{V}$ of Y,

• u-t-p-cliquish (resp. l-t-p-cliquish)[13] at a point $x\in X$ i.e.,

$x\in p{K}_u\left(F\right)$ (resp. $x\in p{K}_l\left(F\right)$), if $x\in \bigcup \left\{Int\left(Cl\left(F^{+}\left(V\right)\right)\right):V\in \mathcal{V}\right\}$

(resp. $x\in \bigcup \left\{Int\left(Cl\left(F^{-}\left(V\right)\right)\right):V\in \mathcal{V}\right\})$ for any open cover $\mathcal{V}$ of Y,

• u-t-β-cliquish (resp. l-t-β-cliquish)[13] at a point $x\in X$ i.e.,

$x\in \beta K_u\left(F\right)$ (resp. $x\in \beta K_l\left(F\right)$), if $x\in \bigcup \left\{Cl\left(Int\left(Cl\left(F^{+}\left(V\right)\right)\right)\right):V\in \mathcal{V}\right\}$

(resp. $x\in \bigcup \left\{Cl\left(Int\left(Cl\left(F^{-}\left(V\right)\right)\right)\right):V\in \mathcal{V}\right\})$ for any open cover $\mathcal{V}$ of Y,

• u-t-cliquish (resp. l-t-cliquish) [55] at a point $x\in X$ i.e.,

$x\in K_u\left(F\right)$ (resp. $x\in K_l\left(F\right)$), if $x\in Cl(\bigcup \left\{Int\left(F^{+}\left(V\right)\right):V\in \mathcal{V}\right\})$

(resp. $x\in Cl(\bigcup \left\{Int\left(F^{-}\left(V\right)\right):V\in \mathcal{V}\right\})$ for any open cover $\mathcal{V}$ of Y,

• pre u-t-cliquish (resp. pre l-t-cliquish) at a point $x\in X$ i.e.,

$x\in P{K}_u\left(F\right)$ (resp. $x\in P{K}_l\left(F\right)$), if $x\in Cl(\bigcup \left\{Int\left(Cl\left(F^{+}\left(V\right)\right)\right):V\in \mathcal{V}\right\})$

(resp. $x\in Cl(\bigcup \left\{Int\left(Cl\left(F^{-}\left(V\right)\right)\right):V\in \mathcal{V}\right\}))$ for any open cover $\mathcal{V}$ of Y.

We have the following characterizations of those types of generalized continuity in terms of the cluster operators.

Lemma 12. 

For any multifunction $F:(X,\tau )\to (Y,\sigma )$ and $x\in X$, the following equivalences hold:

$\left(i\right)$$x\in \alpha .K_u\left(F\right)$(resp. $x\in \alpha .K_l\left(F\right))$ if and only if the family

$〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$(resp. $〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right))$ contains a singleton,

$\left(ii\right)$$x\in q.K_u\left(F\right)$(resp. $x\in q.K_l\left(F\right))$ if and only is the family

$〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$(resp. $〈\mathcal{S},\mathcal{I},\mathcal{S}F〉\left(x\right))$ contains a singleton;

$\left(iii\right)$$x\in p.K_u\left(F\right)$(resp. $x\in p.K_l\left(F\right))$ if and only if the family

$〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$(resp. $〈\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right))$ contains a singleton;

$\left(iv\right)$$x\in \beta .K_u\left(F\right)$(resp. $x\in \beta .K_l\left(F\right))$ if and only if the family

$〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$(resp. $〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(x\right))$ contains a singleton

$\left(v\right)$$x\in K_u\left(F\right)$(resp. $x\in K_l\left(F\right))$ if and only if for any $U\in \pi \left(x\right)$,

the family $l\left(〈\mathcal{I},\mathcal{I}F〉\left(U\right)\right)$ (resp. $l\left(〈\mathcal{I},\mathcal{S}F〉\left(U\right)\right))$ contains a singleton,

$\left(vi\right)$$x\in P{K}_u\left(F\right)$(resp. $x\in P{K}_l\left(F\right))$ if and only if for any $U\in \pi \left(x\right)$,

the family $l\left(〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(U\right)\right)$ (resp. $l\left(〈\mathcal{I},\mathcal{S},\mathcal{S}F〉\left(U\right)\right))$ contains

a singleton.

Proof. 

If $\mathcal{V}$ is an open covering of $(Y,\sigma )$ and $\left\{y\right\}\in 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$

(resp. $\left\{y\right\}\in 〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$, $\left\{y\right\}\in 〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$, $\left\{y\right\}\in 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right))$ for some $y\in Y$, then $\left\{y\right\}\in \mathcal{P}\left(V\right)$ for some $V\in \mathcal{V}$ and according to the equivalences listed in the proof of Remark 9, we obtain

$x\in Int(Cl\left(Int\left(F^{+}\left(V\right)\right)\right)$ (resp. $x\in Cl(Int\left(F^{+}\left(V\right)\right),x\in Int\left(Cl\left(F^{+}\left(V\right)\right)\right),x\in Cl(Int\left(Cl\left(F^{+}\left(V\right)\right)\right)$. So, $x\in \alpha .K_u\left(F\right)$ (resp. $x\in q.K_u\left(F\right)$, $x\in p.K_u\left(F\right)$, $x\in \beta .K_u\left(F\right)$).

Now assume that $x\in \alpha .K_u\left(F\right)$ (resp. $x\in q.K_u\left(F\right)$, $x\in p.K_u\left(F\right)$, $x\in \beta .K_u\left(F\right)$) and $\left\{y\right\}\notin 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$ (resp. $\left\{y\right\}\notin 〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)$, $\left\{y\right\}\notin 〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)$, $\left\{y\right\}\notin 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right))$ for all points $y\in Y$. Then, since the values of cluster functions are closed with respect to the upper Vietoris topology, for every $y\in Y$ there exists an open $V_y\subset Y$ such that

$\left\{y\right\}\in \mathcal{P}\left(V_y\right)$ and $\mathcal{P}\left(V_y\right)\cap 〈\mathcal{I},\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)=\varnothing$

(resp. $\mathcal{P}\left(V_y\right)\cap 〈\mathcal{S},\mathcal{I},\mathcal{I}F〉\left(x\right)=\varnothing$, $\mathcal{P}\left(V_y\right)\cap 〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)=\varnothing$,

$\mathcal{P}\left(V_y\right)\cap 〈\mathcal{S},\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)=\varnothing )$. So, the family $\mathcal{V}=\left\{V_y:y\in Y\right\}$ forms an open covering of Y such that $x\notin \bigcup \left\{Int\left(Cl\left(Int\left(F^{+}\left(V\right)\right)\right)\right):V\in \mathcal{V}\right\}$

(resp. $x\notin \bigcup \left\{Cl\left(Int\left(F^{+}\left(V\right)\right)\right):V\in \mathcal{V}\right\},x\notin \bigcup \left\{Int\left(Cl\left(F^{+}\left(V\right)\right)\right):V\in \mathcal{V}\right\},x\notin \bigcup \left\{Cl\left(Int\left(Cl\left(F^{+}\left(V\right)\right)\right)\right):V\in \mathcal{V}\right\})$ which gives a contradiction and finishes the proof of $\left(i\right)-\left(iv\right)$, the case $"u"$. The proof of the second case is analogous.

To prove $\left(v\right)$ and $\left(vi\right)$, let assume first that $x\in K_u\left(F\right)$

(resp. $x\in P{K}_u\left(F\right)$) and there exists $U\in \pi \left(x\right)$ such that $\left\{y\right\}\notin l\left(〈\mathcal{I},\mathcal{I}F〉\left(U\right)\right)$ (resp. $\left\{y\right\}\notin l\left(〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(U\right)\right)$) for every $y\in Y$. Then, for every $y\in Y$ there exists an open $V_y\subset Y$ such that $\mathcal{P}\left(V_y\right)\cap \overline{\bigcup \left\{〈\mathcal{I},\mathcal{I}F〉\left(x\right):x\in U\right\}}=\varnothing$

(resp. $\mathcal{P}\left(V_y\right)\cap \overline{\bigcup \left\{〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right):x\in U\right\}}=\varnothing$) which is equivalent to

$U\cap Int\left(F^{+}\left(V_y\right)\right)=\varnothing$ (resp. $U\cap Int(Cl\left(F^{+}\left(V_y\right)\right)=\varnothing$. As a result, we obtain an open covering $\mathcal{V}=\left\{V_y:y\in Y\right\}$ of Y such that

$x\notin Cl(\bigcup \left\{Int\left(F^{+}\left(V\right)\right):V\in \mathcal{V}\right\})$ (resp. $x\notin Cl(\bigcup \left\{Int\left(Cl\left(F^{+}\left(V\right)\right)\right):V\in \mathcal{V}\right\}))$ which gives a contradiction.

Now, let us take an open covering $\mathcal{V}$ of $(Y,\sigma )$ and assume that for every $U\in \tau \left(x\right)$ there exists $y\in Y$ such that $\left\{y\right\}\in l\left(〈\mathcal{I},\mathcal{I}F〉\left(U\right)\right)$

(resp. $\left\{y\right\}\in l\left(〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(U\right)\right)$). Then, there exists $V\in \mathcal{V}$ such that $y\in V$ and $\mathcal{P}\left(V\right)\cap \overline{\bigcup \left\{〈\mathcal{I},\mathcal{I}F〉\left(x\right):x\in U\right\}}\ne \varnothing$

(resp. $\mathcal{P}\left(V\right)\cap \overline{\bigcup \left\{〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right):x\in U\right\}}\ne \varnothing$) which means that

$\mathcal{P}\left(V\right)\cap 〈\mathcal{I},\mathcal{I}F〉\left(x\right)\ne \varnothing$ (resp. $\mathcal{P}\left(V\right)\cap 〈\mathcal{I},\mathcal{S},\mathcal{I}F〉\left(x\right)\ne \varnothing$ ) for some $x\in U$.

So, we have $U\cap Int\left(F^{+}\left(V\right)\right)\ne \varnothing$ (resp. $U\cap Int(Cl\left(F^{+}\left(V\right)\right)\ne \varnothing$.

This shows that $U\cap \bigcup \left\{Int\left(F^{+}\left(V\right)\right):V\in \mathcal{V}\right\}\ne \varnothing$

(resp. $U\cap \bigcup \left\{Int\left(Cl\left(F^{+}\left(V\right)\right)\right):V\in \mathcal{V}\right\}\ne \varnothing )$ for any open set U containing x, i.e., that $x\in Cl(\bigcup \left\{Int\left(F^{+}\left(V\right)\right):V\in \mathcal{V}\right\})$

(resp. $x\in Cl(\bigcup \left\{Int\left(Cl\left(F^{+}\left(V\right)\right)\right):V\in \mathcal{V}\right\}))$ and finishes the proof of $\left(v\right)$ and $\left(vi\right)$, the case $"u"$. The proof of the second case is analogous. □