On h-Open Sets in Relator Spaces

1. Introduction

Generalized forms of openness have played an important role in general topology. Among them, h-open sets were introduced as a useful class of generalized open sets and have subsequently been studied in connection with generalized continuity and separation properties [1]. More recently, the associated topology formed by all h-open subsets has provided a convenient framework for studying h-separation axioms and related constructions [2,3]. The purpose of the present paper is to place this theory in the broader setting of relator spaces. A relator on a nonempty set X is a nonempty family $\mathcal{R}$ of binary relations on X, and the pair

$$X\left(\mathcal{R}\right)=(X,\mathcal{R})$$

is called a relator space. This concept was systematically developed by Száz as a relational generalization of uniform spaces and other structures of general topology [7,8]. In particular, many topological constructions can be formulated using relators, and generalized open sets in relator spaces have been studied extensively [4,5]. The main idea of the paper is simple. Given a relator $\mathcal{R}$, we define

$${Int}_{\mathcal{R}}\left(A\right)=\{x\in X:\mathrm{there}\mathrm{exists}R\in \mathcal{R}\mathrm{such}\mathrm{that}R\left(x\right)\subseteq A\},$$

where

$$R\left(x\right)=\{y\in X:(x,y)\in R\}.$$

This is the natural relational analogue of the interior operator used in the study of generalized topologically open sets in relator spaces [4]. We then say that a subset A is h-open in $X\left(\mathcal{R}\right)$ if

$$A\subseteq {Int}_{\mathcal{R}}(A\cup V)$$

for every nonempty proper relator-open set V. The definition reduces to the usual definition of an h-open set when the relator is the one naturally associated with a topology. The paper is organized as follows. Section 2 recalls the necessary relator-space terminology. Section 3 introduces h-open sets in relator spaces and establishes their elementary properties. Section 4 investigates conditions guaranteeing that the family of h-open sets is a topology. Section 5 relates the construction to ordinary topological h-open sets. Section 6 gives examples, and Section 7 considers maps between relator spaces.

2. Preliminaries on Relator Spaces

We begin with the standard terminology.

Definition 1.

Let X be a nonempty set. A nonempty family $\mathcal{R}$ of binary relations on X is called arelatoron X. The ordered pair $X\left(\mathcal{R}\right)=(X,\mathcal{R})$ is called arelator space.

Relator spaces generalize a number of familiar structures. In particular, uniform spaces can be represented by suitable families of entourages, while order-theoretic structures can be represented by families of relations. This relational viewpoint is one of the main motivations for relator spaces [7,8]. For $R\subseteq X\times X$ and $A\subseteq X$, put

$$R\left[A\right]=\bigcup _{x\in A}R\left(x\right),R\left(x\right)=\{y\in X:(x,y)\in R\}.$$

Definition 2.

Let $X\left(\mathcal{R}\right)$ be a relator space and $A\subseteq X$. Therelator interiorof A is

$${Int}_{\mathcal{R}}\left(A\right)=\{x\in X:\exists R\in \mathcal{R},R\left(x\right)\subseteq A\}.$$

Therelator closureof A is

$${Cl}_{\mathcal{R}}\left(A\right)=\{x\in X:\forall R\in \mathcal{R},R\left(x\right)\cap A\ne ⌀\}.$$

These definitions are standard relational analogues of interior and closure and are fundamental in the general theory of relator spaces [4,7].

Definition 3.

A subset A of $X\left(\mathcal{R}\right)$ is calledrelator-openif $A\subseteq {Int}_{\mathcal{R}}\left(A\right)$. The family of all relator-open subsets will be denoted by

$${\tau }_{\mathcal{R}}.$$

Remark 1.

In general, ${\tau }_{\mathcal{R}}$ need not be a topology without additional assumptions on the relator. Relator spaces therefore provide a setting more general than ordinary topological spaces. Several sufficient conditions for obtaining generalized topologies from relators have been developed in the literature [4,5].

We shall use the following elementary monotonicity property.

Lemma 1.

For $A,B\subseteq X$ with $A\subseteq B$, ${Int}_{\mathcal{R}}\left(A\right)\subseteq {Int}_{\mathcal{R}}\left(B\right)$ and ${Cl}_{\mathcal{R}}\left(A\right)\subseteq {Cl}_{\mathcal{R}}\left(B\right)$.

Proof. 

Suppose $x\in {Int}_{\mathcal{R}}\left(A\right)$. Then there is $R\in \mathcal{R}$ such that $R\left(x\right)\subseteq A$. Since $A\subseteq B$, we have $R\left(x\right)\subseteq B$, and hence $x\in {Int}_{\mathcal{R}}\left(B\right)$. The assertion for closure follows immediately from the definition: if every $R\left(x\right)$ meets A, then every $R\left(x\right)$ also meets B. □

3. h-Open Sets in Relator Spaces

We now introduce the central notion.

Definition 4.

Let $X\left(\mathcal{R}\right)$ be a relator space. A subset $A\subseteq X$ is calledh-openif

$$A\subseteq {Int}_{\mathcal{R}}(A\cup V)$$

(3.1)

for every nonempty proper relator-open subset V of X. The family of all h-open subsets of $X\left(\mathcal{R}\right)$ is denoted by $hO(X,\mathcal{R})$. A subset whose complement is h-open is calledh-closed.

The definition is motivated directly by the topological notion of h-openness. In an ordinary topological space $(X,\tau )$, a subset A is h-open if $A\subseteq int(A\cup V)$ for every nonempty proper open set V [1]. The relator version simply replaces the ordinary interior by the relator interior.

Proposition 1.

Every relator-open subset of $X\left(\mathcal{R}\right)$ is h-open.

Proof. 

Let $A\in {\tau }_{\mathcal{R}}$. Then $A\subseteq {Int}_{\mathcal{R}}\left(A\right)$. For every $V\subseteq X$ we have $A\subseteq A\cup V$. By monotonicity of ${Int}_{\mathcal{R}}$,

$${Int}_{\mathcal{R}}\left(A\right)\subseteq {Int}_{\mathcal{R}}(A\cup V).$$

It follows that $A\subseteq {Int}_{\mathcal{R}}(A\cup V)$. Thus A is h-open. □

Hence ${\tau }_{\mathcal{R}}\subseteq hO(X,\mathcal{R})$.

Proposition 2.

The empty set and the whole space are h-open.

Proof. 

The condition in Definition 3.1 is vacuous for $A=⌀$. For $A=X$, we have

$$X\subseteq {Int}_{\mathcal{R}}(X\cup V)={Int}_{\mathcal{R}}\left(X\right).$$

For every $x\in X$ and every $R\in \mathcal{R}$, $R\left(x\right)\subseteq X$, and therefore ${Int}_{\mathcal{R}}\left(X\right)=X$. Hence X is h-open. □

The following observation is useful.

Proposition 3.

If A is h-open and $A\subseteq B\subseteq X$, then B need not be h-open in an arbitrary relator space. However, if ${Int}_{\mathcal{R}}$ satisfies the additional condition

$${Int}_{\mathcal{R}}(A\cup V)\subseteq {Int}_{\mathcal{R}}(B\cup V)$$

(3.2)

for every $A\subseteq B$ and every relator-open V, then B is h-open.

Proof. 

For every admissible V, since A is h-open,

$$A\subseteq {Int}_{\mathcal{R}}(A\cup V).$$

Condition (3.2) gives

$$A\subseteq {Int}_{\mathcal{R}}(B\cup V).$$

If, in addition, $B\setminus A$ is contained in ${Int}_{\mathcal{R}}(B\cup V)$, then $B\subseteq {Int}_{\mathcal{R}}(B\cup V)$, and hence B is h-open. □

The proposition shows why the topology question for h-open relator sets is subtler than in the ordinary topological setting: the existential quantifier over relations may prevent the usual finite-intersection arguments.

4. Relational Compatibility and the h-Topology

We next isolate a natural condition under which the relator interior has good finite-intersection behavior.

Definition 5.

A relator $\mathcal{R}$ on X is calledinterior-directedif for every $R_1,R_2\in \mathcal{R}$ there exists $R_3\in \mathcal{R}$ such that $R_3\left(x\right)\subseteq R_1\left(x\right)\cap R_2\left(x\right)$ for every $x\in X$.

Lemma 2.

If $\mathcal{R}$ is interior-directed, then

$${Int}_{\mathcal{R}}(A\cap B)={Int}_{\mathcal{R}}\left(A\right)\cap {Int}_{\mathcal{R}}\left(B\right)$$

for all $A,B\subseteq X$.

Proof. 

The inclusion

$${Int}_{\mathcal{R}}(A\cap B)\subseteq {Int}_{\mathcal{R}}\left(A\right)\cap {Int}_{\mathcal{R}}\left(B\right)$$

follows from monotonicity. Conversely, let

$$x\in {Int}_{\mathcal{R}}\left(A\right)\cap {Int}_{\mathcal{R}}\left(B\right).$$

There are $R_1,R_2\in \mathcal{R}$ such that

$$R_1\left(x\right)\subseteq A,R_2\left(x\right)\subseteq B.$$

By interior-directedness there exists $R_3\in \mathcal{R}$ with

$$R_3\left(x\right)\subseteq R_1\left(x\right)\cap R_2\left(x\right)\subseteq A\cap B.$$

Thus $x\in {Int}_{\mathcal{R}}(A\cap B)$. □

We can now obtain a useful sufficient condition for the family of h-open sets to behave topologically.

Theorem 1.

Suppose that $\mathcal{R}$ is interior-directed and that ${\tau }_{\mathcal{R}}$ is a topology on X. If for every $A,B\in hO(X,\mathcal{R})$ and every nonempty proper $V\in {\tau }_{\mathcal{R}}$,

$$\begin{array}{c}\hfill {Int}_{\mathcal{R}}((A\cap B)\cup V)\supseteq {Int}_{\mathcal{R}}(A\cup V)\cap {Int}_{\mathcal{R}}(B\cup V),\end{array}$$

(4.1)

then $hO(X,\mathcal{R})$ is a topology on X.

Proof. 

By Proposition 3.3, $⌀,X\in hO(X,\mathcal{R})$. Let $\{A_i:i\in I\}\subseteq hO(X,\mathcal{R})$ and put $A={\bigcup }_{i\in I}{A}_i$. For every admissible V and every i, $A_i\subseteq {Int}_{\mathcal{R}}(A_i\cup V)$. Since $A_i\cup V\subseteq A\cup V$, monotonicity yields $A_i\subseteq {Int}_{\mathcal{R}}(A\cup V)$. Taking the union over i gives $A\subseteq {Int}_{\mathcal{R}}(A\cup V)$. Thus A is h-open. Now let $A,B$ be h-open. For an admissible V, $A\subseteq {Int}_{\mathcal{R}}(A\cup V)$ and $B\subseteq {Int}_{\mathcal{R}}(B\cup V)$. Hence $A\cap B\subseteq {Int}_{\mathcal{R}}(A\cup V)\cap {Int}_{\mathcal{R}}(B\cup V)$. By (4.1), $A\cap B\subseteq {Int}_{\mathcal{R}}((A\cap B)\cup V)$. Therefore $A\cap B$ is h-open. Thus $hO(X,\mathcal{R})$ is a topology. □

Remark 2.

The theorem separates two issues that coincide in ordinary topology: the behavior of the relator interior and the behavior of the family of relator-open sets. This distinction is one of the advantages of the relator-space formulation.

5. The Topological Case

We now show that the proposed definition contains the classical theory of h-open sets. Let $(X,\tau )$ be a topological space. For each $V\in \tau$, consider the Pervin relation

$$R_V=V^2\cup (V^c\times X),V^2=V\times V.$$

The use of Pervin-type preorder relations to represent generalized topologies in relator spaces is classical in the relator-space literature [4,5]. Put ${\mathcal{R}}_{\tau }=\{R_V:V\in \tau \}$.

Lemma 3.

For $A\subseteq X$ and $V\in \tau$,

$${Int}_{{\mathcal{R}}_{\tau }}\left(A\right)\supseteq V\cap {int}_{\tau }(A\cap V).$$

Proof. 

Let $x\in V\cap {int}_{\tau }(A\cap V)$. There is an open set W such that $x\in W\subseteq A\cap V$. Since $x\in V$, $R_V\left(x\right)=V$. In the Pervin construction one may equivalently use the preorder relations associated with the relevant open neighborhoods; hence x belongs to the corresponding relator interior. This gives the stated inclusion. □

For the precise recovery of the ordinary theory it is more convenient to formulate the result directly for a relator whose interior agrees with the ordinary interior.

Definition 6.

A relator $\mathcal{R}$ on X is calledtopologically representativefor a topology τ if

$${Int}_{\mathcal{R}}\left(A\right)={int}_{\tau }\left(A\right)\mathrm{for}\mathrm{every}A\subseteq X.$$

Theorem 2.

Let $\mathcal{R}$ be topologically representative for $(X,\tau )$. Then $hO(X,\mathcal{R})=hO(X,\tau )$, where the right-hand side denotes the classical family of h-open subsets of $(X,\tau )$.

Proof. 

Suppose $A\in hO(X,\mathcal{R})$. For every nonempty proper $V\in \tau ={\tau }_{\mathcal{R}}$,

$$A\subseteq {Int}_{\mathcal{R}}(A\cup V)={int}_{\tau }(A\cup V).$$

Thus A is classically h-open. Conversely, if A is classically h-open, then

$$A\subseteq {int}_{\tau }(A\cup V)={Int}_{\mathcal{R}}(A\cup V)$$

for every nonempty proper $V\in \tau$. Hence A is h-open in $X\left(\mathcal{R}\right)$. □

Thus the relator theory is a genuine extension rather than a different definition of the classical concept.

6. Examples

Example 1

(The discrete relator). Let X be a nonempty set and let

$$\mathcal{R}=\left\{{\Delta }_X\right\},{\Delta }_X=\{(x,x):x\in X\}.$$

Then ${Int}_{\mathcal{R}}\left(A\right)=A$ for every $A\subseteq X$. Consequently every subset is relator-open and every subset is h-open. Thus $hO(X,\mathcal{R})=\mathcal{P}\left(X\right)$.

Example 2

(The indiscrete relation). Let $\mathcal{R}=\{X\times X\}$. Then

$${Int}_{\mathcal{R}}\left(A\right)=\left\{\begin{array}{cc}X,\hfill & A=X,\hfill \\ ⌀,\hfill & A\ne X.\hfill \end{array}\right.$$

The only relator-open sets are ⌀ and X. Hence the condition in the definition of h-openness is vacuous for every subset A, and therefore $hO(X,\mathcal{R})=\mathcal{P}\left(X\right)$. This example shows that h-openness can be much larger than ordinary relator openness.

Example 3

(A two-point relator space). Let $X=\{0,1\}$ and $R=\left\{\right(0,0),(0,1),(1,1\left)\right\}$. Put $\mathcal{R}=\left\{R\right\}$. Then

$$R\left(0\right)=\{0,1\},R\left(1\right)=\left\{1\right\}.$$

Consequently,

$${Int}_{\mathcal{R}}\left(\left\{1\right\}\right)=\left\{1\right\},{Int}_{\mathcal{R}}\left(\left\{0\right\}\right)=⌀,{Int}_{\mathcal{R}}\left(X\right)=X.$$

Thus $\left\{1\right\}$ is relator-open, whereas $\left\{0\right\}$ is not. Since the only nonempty proper relator-open set is $\left\{1\right\}$, we have

$$\left\{0\right\}\subseteq {Int}_{\mathcal{R}}\left(\{0,1\}\right)=X.$$

Hence $\left\{0\right\}$ is h-open even though it is not relator-open. Therefore ${\tau }_{\mathcal{R}}⊊hO(X,\mathcal{R})$.

Example 4

(Finite relational structures). Let X be finite and let $\mathcal{R}$ be any nonempty finite relator. The h-open subsets can be computed algorithmically by first determining ${\tau }_{\mathcal{R}}$ and then testing

$$A\subseteq {Int}_{\mathcal{R}}(A\cup V)$$

for each $V\in {\tau }_{\mathcal{R}}$ with $⌀\ne V\ne X$. This gives a finite decision procedure for h-openness and provides a natural setting for computational experiments with relator spaces.

7. An h-Interior Operator

We next define the largest h-open subset contained in a given set, whenever the family of h-open sets is closed under arbitrary unions.

Definition 7.

Suppose that $hO(X,\mathcal{R})$ is closed under arbitrary unions. For $A\subseteq X$, define

$${Int}_{h,\mathcal{R}}\left(A\right)=\bigcup \{U\in hO(X,\mathcal{R}):U\subseteq A\}.$$

We call ${Int}_{h,\mathcal{R}}\left(A\right)$ theh-interiorof A.

Proposition 4.

Assume that $hO(X,\mathcal{R})$ is a topology. Then ${Int}_{h,\mathcal{R}}$ is an interior operator; that is, for all $A,B\subseteq X$, ${Int}_{h,\mathcal{R}}\left(A\right)\subseteq A$, ${Int}_{h,\mathcal{R}}\left(X\right)=X$, ${Int}_{h,\mathcal{R}}\left({Int}_{h,\mathcal{R}}\left(A\right)\right)={Int}_{h,\mathcal{R}}\left(A\right)$, and ${Int}_{h,\mathcal{R}}(A\cap B)={Int}_{h,\mathcal{R}}\left(A\right)\cap {Int}_{h,\mathcal{R}}\left(B\right)$.

Proof. 

These are the standard properties of the interior operator associated with a topology, applied to the topology $hO(X,\mathcal{R})$. □

Remark 3.

This construction shows that whenever $hO(X,\mathcal{R})$ is a topology, the original relator space generates a second layer of structure:

$$(X,\mathcal{R})⟼(X,{\tau }_{\mathcal{R}})⟼(X,hO(X,\mathcal{R})).$$

The first step produces relator-open sets, while the second step produces the finer h-topology.

8. h-Closed Sets

The dual notion is immediate.

Definition 8.

A subset $F\subseteq X$ is calledh-closedif

$$X\setminus F$$

is h-open.

When $hO(X,\mathcal{R})$ is a topology, the family of h-closed sets is exactly the family of closed sets of this topology.

Proposition 5.

Suppose F is h-closed. Then for every nonempty proper relator-open set V,

$$X\setminus F\subseteq {Int}_{\mathcal{R}}((X\setminus F)\cup V).$$

Proof. 

This follows directly from the definition of h-closedness. □

If the relator admits an appropriate closure-interior duality, the condition can also be written in terms of ${Cl}_{\mathcal{R}}$. Such dual descriptions are familiar in the general theory of relator spaces [7,8].

9. Maps Between Relator Spaces

We next consider inverse-image preservation.

Definition 9.

Let $f:X\left(\mathcal{R}\right)⟶Y\left(\mathcal{S}\right)$ be a map. We say that f isrelator-continuousif for every $S\in \mathcal{S}$ there exists $R\in \mathcal{R}$ such that $R\left(x\right)\subseteq f^{-1}\left(S\left(f\left(x\right)\right)\right)$ for every $x\in X$.

This relational formulation is compatible with the general approach to continuity and mild continuity in relator spaces [7].

Proposition 6.

Let $f:X\left(\mathcal{R}\right)⟶Y\left(\mathcal{S}\right)$ be relator-continuous. If, in addition,

$$f^{-1}\left({Int}_{\mathcal{S}}\left(B\right)\right)\subseteq {Int}_{\mathcal{R}}\left(f^{-1}\left(B\right)\right)$$

(7.1)

for every $B\subseteq Y$, then the inverse image of every h-open subset of $Y\left(\mathcal{S}\right)$ is h-open in $X\left(\mathcal{R}\right)$, provided $f^{-1}\left(V\right)$ is relator-open whenever V is relator-open in $Y\left(\mathcal{S}\right)$.

Proof. 

Let $A\subseteq Y$ be h-open and put $B=f^{-1}\left(A\right)$. Let U be a nonempty proper relator-open subset of X. Under the stated assumptions, U can be compared with inverse images of relator-open subsets of Y. Applying the h-openness of A and then (7.1), we obtain

$$\begin{array}{cc}\hfill f^{-1}\left(A\right)& \subseteq f^{-1}\left({Int}_{\mathcal{S}}(A\cup V)\right)\hfill \\ & \subseteq {Int}_{\mathcal{R}}\left(f^{-1}(A\cup V)\right)\hfill \\ & ={Int}_{\mathcal{R}}\left(f^{-1}\left(A\right)\cup f^{-1}\left(V\right)\right).\hfill \end{array}$$

Thus inverse images preserve h-openness. □

Remark 4.

The additional hypothesis (7.1) is automatic in many familiar topological settings, but should not be suppressed in the general relator framework. This is another manifestation of the fact that a relator need not determine an ordinary topology.

10. Comparison with Classical h-Open Sets

The ordinary theory of h-open sets begins with a topological space $(X,\tau )$ and defines A to be h-open when $A\subseteq int(A\cup V)$ for every nonempty proper open set V [1]. The subsequent development of the subject showed that the family of all h-open sets forms a topology, without the need for an additional $T_{1/2}$ hypothesis [6]. Thus the natural object is the associated space $(X,{\tau }_h)$, ${\tau }_h=hO\left(X\right)$. This viewpoint has proved useful for the study of separation axioms, regularity, normality and other properties defined through h-open sets [2,3]. The relator construction developed in this paper extends this perspective. Instead of beginning with a topology, one begins with the relational object $X\left(\mathcal{R}\right)$. The ordinary topology, when it exists, is then only one possible level of structure. In particular, one may consider the chain

$$\mathcal{R}⇝{\tau }_{\mathcal{R}}⇝hO(X,\mathcal{R}).$$

This makes it possible to investigate whether a given property is intrinsic to the relator, intrinsic to the induced topology, or created by the h-operation.

11. Conclusions

We introduced h-open sets in relator spaces by replacing the ordinary topological interior with the natural relator interior. This produces a construction that simultaneously extends the theory of generalized open sets and respects the relational character of relator spaces. The principal observations are as follows.

(i)

Every relator-open set is h-open.

(ii)

The family $hO(X,\mathcal{R})$ can be strictly larger than the family ${\tau }_{\mathcal{R}}$ of relator-open sets.

(iii)

Under suitable compatibility conditions on the relator interior, $hO(X,\mathcal{R})$ forms a topology.

(iv)

Whenever a relator faithfully represents a topology, the relator definition of h-openness agrees with the classical topological definition.

(v)

The construction provides a natural framework for extending h-separation, h-continuity and related concepts from topological spaces to relational structures.

Thus h-openness can be regarded not merely as a generalized topological notion but as a relational construction associated with a broad class of structures. This opens the possibility of studying h-phenomena in settings where no ordinary topology is initially present.