Weakly h-Continuous Functions via h-Open Sets

1. Introduction

Weak continuity was introduced by Levine [13] and was subsequently studied and compared with continuity and almost continuity by Noiri [17,18], Rose [27], and Neubrunn [16]. The broader programme of replacing ordinary neighbourhoods by generalized open sets includes semi-continuity [14], almost continuity [30], weak precontinuity [15], weak $\alpha$-continuity [20], weak b-continuity [29], and almost weak continuity [24]. Almost weak continuity was subsequently developed for multifunctions by Popa and Noiri [25], while Noiri’s $\theta$-precontinuous functions provide another closure-based weak form of continuity [21]. The classical closed-graph theory for weakly continuous functions was developed by Noiri [19], and closely related $\theta$-type closure and separation constructions occur in the work of Janković [5] and Veličko [31].

A particularly relevant precursor is the minimal-structure theory of Popa and Noiri. They introduced weakly m-continuous functions and studied their relations with strongly m-closed graphs, m-compactness and m-connectedness [26]. Noiri and Popa subsequently introduced $(i,j)$-weakly m-continuous functions for bitopological codomains and obtained unified characterizations, separation, compact-image and discontinuity-set results [23]. This general framework contains the basic notion studied here: taking

$$m_X={\tau }_h\left(X\right),{\sigma }_1=\sigma ,{\sigma }_2={\sigma }_h\left(Y\right),$$

their $(1,2)$-weakly m-continuity is exactly weak h-continuity. Thus the results below that depend only on this minimal/bitopological structure are recorded as specializations of that theory, while the associated-topology interpretation, the h-specific examples, compatibility issues and the fully associated weakly h-irresolute theory are treated in their own right.

A recent development close in spirit to the present paper is due to Almuhur, Al-Labadi and Bashir [2], who introduced $h\alpha$-almost weakly continuous functions in ideal topological spaces. Their framework is genuinely different from ours: it is built from $h\alpha$-open sets and the corresponding ideal-based interior and closure operators, whereas weak h-continuity here is formulated using h-open neighbourhoods in the domain and ordinary open neighbourhoods in the codomain together with h-closure. Nevertheless, the two approaches belong to the same broader programme of weakening continuity through generalized open sets, and several separation phenomena have closely parallel forms. The present paper studies the corresponding construction based specifically on h-open sets and the associated topology.

The notion of an h-open set originates with Abbas [1]; subsequent corrections and developments appear in [3,28]. A decisive structural point is that the family of all h-open subsets of an arbitrary topological space is itself a topology; related topological viewpoints also appear in [4]. We denote this associated topology by ${\tau }_h\left(X\right)$. Recent work has used the associated space systematically for separation axioms, regularity and normality, compatible subspaces and products, sequential separation, and h-connected or hyperconnected structures; see [6,7,8,9,11,12].

This viewpoint is particularly useful here, but it must be applied with care. The definition of weak h-continuity uses ordinary open neighbourhoods in the codomain and their h-closures. Hence it is not, in general, simply classical weak continuity between two associated spaces. The distinction is responsible for several corrections to the naive transfer of classical arguments. In particular, closure identities involving products require an explicit compatibility hypothesis, and an h-compact subset of an ambient space must be distinguished from an intrinsically h-compact subspace unless the associated topology commutes with formation of that subspace.

The paper is organized as follows. Section 2 recalls the associated topology and h-$\theta$-closure. Section 3 develops weak h-continuity, including finite examples and the specialization of the general minimal-structure characterizations. Section 4 introduces weak h-irresoluteness as the fully associated version of weak continuity. Section 5 treats graph properties and product compatibility. Section 6 studies images of relatively h-compact sets, and Section 7 gives a frontier characterization of the failure of weak h-continuity.

2. Preliminaries and the Associated Topology

Throughout the paper, $(X,\tau )$ and $(Y,\sigma )$ denote topological spaces, with no separation assumptions unless explicitly stated. If the topology is clear, ordinary closure and interior are denoted by Cl and Int.

Definition 1

([1]). A subset $A\subseteq X$ ish-openif

$$A\subseteq {Int}_{\tau }(A\cup U)$$

for every nonempty proper open set $U\in \tau$. We write $hO\left(X\right)$ for the family of all h-open subsets of X.

The following structural fact is fundamental. In the generality used here it was established by Sharma, Saproo, Billawria and Digra [28]; see also the associated-topology viewpoint of [4].

Theorem 1.

For every topological space $(X,\tau )$, the family

$${\tau }_h\left(X\right):=hO\left(X\right)$$

is a topology on X, and $\tau \subseteq {\tau }_h\left(X\right)$.

We call $(X,{\tau }_h\left(X\right))$ the associated space.

Definition 2.

A topological space $(X,\tau )$ is calledh-fixedif its associated topology coincides with its original topology, that is,

$${\tau }_h\left(X\right)=\tau .$$

Equivalently, every h-open subset of X is open. Thus, in an h-fixed space, the h-open sets are precisely the open sets, the h-closed sets are precisely the closed sets, and the associated space coincides with the original space.

Henceforth

$${hCl}_X\left(A\right):={Cl}_{{\tau }_h\left(X\right)}\left(A\right),{hInt}_X\left(A\right):={Int}_{{\tau }_h\left(X\right)}\left(A\right).$$

Subscripts are omitted when no ambiguity is possible.

Since ${hCl}_X$ is ordinary closure in the associated space, its standard neighbourhood characterization will be used repeatedly.

Lemma 1.

For $A\subseteq X$ and $x\in X$,

$$x\in {hCl}_X\left(A\right)⟺U\cap A\ne ⌀\mathrm{for}\mathrm{every}U\in {\tau }_h\left(X\right)\mathrm{with}x\in U.$$

Proof. 

By definition, ${hCl}_X\left(A\right)$ is the ordinary closure of A in $(X,{\tau }_h\left(X\right))$. The assertion is therefore the usual neighbourhood characterization of closure in that topology. □

The following operator is the associated-topology analogue of the classical $\theta$-closure construction; compare [5,31].

Definition 3.

A point $x\in X$ is anh-$\theta$-cluster pointof $A\subseteq X$ if

$${hCl}_X\left(U\right)\cap A\ne ⌀$$

for every $U\in {\tau }_h\left(X\right)$ with $x\in U$. The set of all such points is denoted by ${hCl}_{\theta }\left(A\right)$. A set A ish-$\theta$-closedif $A={hCl}_{\theta }\left(A\right)$.

The corresponding classical $\theta$-closure fact is standard; see [5,31]. In the h-setting the same argument is read in the associated topology, and the general inclusion ${hCl}_X\left(A\right)\subseteq {hCl}_{\theta }\left(A\right)$ is also recorded in [12, Lemma 2]. We shall need the following elementary consequence for h-open sets.

Proposition 1.

If $A\in {\tau }_h\left(X\right)$, then

$${hCl}_X\left(A\right)={hCl}_{\theta }\left(A\right).$$

Proof. 

The inclusion ${hCl}_X\left(A\right)\subseteq {hCl}_{\theta }\left(A\right)$ holds for every $A\subseteq X$. For the reverse inclusion, let $x\notin {hCl}_X\left(A\right)$ and put

$$U:=X\setminus {hCl}_X\left(A\right).$$

Then $U\in {\tau }_h\left(X\right)$ and $x\in U$. Moreover ${hCl}_X\left(U\right)\cap A=⌀$: indeed, if $a\in A$, then $A\cap U=⌀$, and the h-open neighbourhood A of a is therefore disjoint from U; hence $a\notin {hCl}_X\left(U\right)$ by Lemma 1. Thus $x\notin {hCl}_{\theta }\left(A\right)$, proving ${hCl}_{\theta }\left(A\right)\subseteq {hCl}_X\left(A\right)$. □

Definition 4

([1,28]). A function $f:(X,\tau )\to (Y,\sigma )$ is

(i)

h-continuousif $f^{-1}\left(V\right)\in {\tau }_h\left(X\right)$ for every $V\in \sigma$;

(ii)

h-irresoluteif $f^{-1}\left(V\right)\in {\tau }_h\left(X\right)$ for every $V\in {\sigma }_h\left(Y\right)$.

Equivalently, h-continuity is ordinary continuity $(X,{\tau }_h\left(X\right))\to (Y,\sigma )$, whereas h-irresoluteness is ordinary continuity $(X,{\tau }_h\left(X\right))\to (Y,{\sigma }_h\left(Y\right))$.

The Hausdorff fixed-point property of the associated topology was proved in [10, Proposition 2.6]. Since it is used repeatedly below, we recall only its statement.

Proposition 2

(Hausdorff rigidity, cf. [10]). If $(Y,\sigma )$ is Hausdorff, then Y is h-fixed; equivalently, ${\sigma }_h\left(Y\right)=\sigma$.

Definition 5

([8,9,11]). A subset $A\subseteq X$ ish-subspace compatibleif

$${\tau }_h\left(A\right)={\tau }_h{\left(X\right)|}_A,$$

where ${\tau }_h\left(A\right)$ is formed from the original subspace topology ${\tau |}_A$. A pair $(X,\tau ),(Y,\sigma )$ ish-product compatibleif

$${(\tau \times \sigma )}_h(X\times Y)={\tau }_h\left(X\right)\times {\sigma }_h\left(Y\right).$$

These equalities are hypotheses, not automatic identities in general; this is precisely why subspace and product arguments below state compatibility whenever it is needed.

3. Weakly H-Continuous Functions

Definition 6.

A function $f:(X,\tau )\to (Y,\sigma )$ isweakly h-continuousat $x\in X$ if, for every $V\in \sigma$ with $f\left(x\right)\in V$, there exists $U\in {\tau }_h\left(X\right)$ with $x\in U$ such that

$$f\left(U\right)\subseteq {hCl}_Y\left(V\right).$$

It isweakly h-continuousif it has this property at every point.

With $m_X={\tau }_h\left(X\right)$ and with Y regarded as the bitopological space $(Y,\sigma ,{\sigma }_h\left(Y\right))$, Definition 6 is exactly the $(1,2)$-weakly m-continuous notion of Noiri and Popa [23, Definition 3.5]. The earlier one-topology notion of weak m-continuity of Popa and Noiri [26] is recovered when the same codomain topology is used both for neighbourhoods and closures. Since $\sigma \subseteq {\sigma }_h\left(Y\right)$, weak h-continuity therefore implies the corresponding weak m-continuity for $m_X={\tau }_h\left(X\right)$ and codomain $(Y,\sigma )$.

Every h-continuous function is weakly h-continuous. The converse fails, even for finite spaces.

Example 1

(Weakly h-continuous but not h-continuous). Let

$$X=Y=\{a,b,c\},\tau =\{⌀,\{a\},X\},$$

and

$$\sigma =\{⌀,\{a\},\{a,b\},Y\}.$$

A direct computation from the definition gives

$${\tau }_h\left(X\right)=\{⌀,\left\{a\right\},\{b,c\},X\}$$

and

$${\sigma }_h\left(Y\right)=\{⌀,\left\{a\right\},\left\{b\right\},\{a,b\},\{b,c\},Y\}.$$

Let $f:X\to Y$ be the identity function. In $(Y,{\sigma }_h\left(Y\right))$,

$${hCl}_Y\left(\left\{a\right\}\right)=\left\{a\right\},{hCl}_Y\left(\{a,b\}\right)=Y.$$

At a, for the open neighbourhood $\left\{a\right\}$ one may take the h-open neighbourhood $\left\{a\right\}$; for $\{a,b\}$ its h-closure is all of Y. At b the only proper ordinary open neighbourhood is $\{a,b\}$, again with h-closure Y, and at c the only ordinary open neighbourhood is Y. Thus f is weakly h-continuous. On the other hand, $f^{-1}\left(\{a,b\}\right)=\{a,b\}\notin {\tau }_h\left(X\right)$, so f is not h-continuous.

The next statement identifies exactly how the associated topology enters the definition.

Proposition 3.

If $f:(X,\tau )\to (Y,\sigma )$ is weakly h-continuous, then

$$f:(X,{\tau }_h\left(X\right))⟶(Y,\sigma )$$

is weakly continuous in the classical sense of Levine [13]. If Y is h-fixed, the converse also holds.

Proof. 

Because $\sigma \subseteq {\sigma }_h\left(Y\right)$, closure in the finer topology is smaller:

$${hCl}_Y\left(V\right)\subseteq {Cl}_{\sigma }\left(V\right)(V\subseteq Y).$$

Hence the defining inclusion for weak h-continuity implies the classical weak-continuity inclusion. If Y is h-fixed, then ${\sigma }_h\left(Y\right)=\sigma$, so the two closure operators coincide and the two definitions agree. □

In particular, Proposition 2 yields equivalence whenever the codomain is Hausdorff. The h-fixed hypothesis cannot be omitted.

Example 2.

Let $X=\{a,b,c\}$ with $\tau =\{⌀,\{a\},X\}$, let $Y=\{0,1\}$ with $\sigma =\{⌀,\{0\},Y\}$, and define

$$f\left(a\right)=f\left(b\right)=0,f\left(c\right)=1.$$

Then

$${\tau }_h\left(X\right)=\{⌀,\left\{a\right\},\{b,c\},X\},{\sigma }_h\left(Y\right)=\mathcal{P}\left(Y\right).$$

The function $f:(X,{\tau }_h\left(X\right))\to (Y,\sigma )$ is classically weakly continuous, because ${Cl}_{\sigma }\left(\left\{0\right\}\right)=Y$. However, ${hCl}_Y\left(\left\{0\right\}\right)=\left\{0\right\}$. At b, every h-open neighbourhood contains c or equals X, and therefore its image is not contained in $\left\{0\right\}$. Consequently f is not weakly h-continuous.

The basic criteria for weak h-continuity fit naturally into the minimal-structure framework of Noiri and Popa [23]. We record the resulting equivalent formulations in h-notation, since they will be used repeatedly below; the symbols ${Int}_{\sigma }$ and ${Cl}_{\sigma }$ indicate the places where the original codomain topology remains essential.

Theorem 2.

For a function $f:(X,\tau )\to (Y,\sigma )$, the following are equivalent:

(i)

f is weakly h-continuous;

(ii)

for every $V\in \sigma$,

$$f^{-1}\left(V\right)\subseteq {hInt}_X\left(f^{-1}\left({hCl}_Y\left(V\right)\right)\right);$$

(iii)

for every σ-closed set $F\subseteq Y$,

$${hCl}_X\left(f^{-1}\left({hInt}_Y\left(F\right)\right)\right)\subseteq f^{-1}\left(F\right);$$

(iv)

for every $B\subseteq Y$,

$${hCl}_X\left(f^{-1}\left({hInt}_Y\left({Cl}_{\sigma }B\right)\right)\right)\subseteq f^{-1}\left({Cl}_{\sigma }B\right);$$

(v)

for every $B\subseteq Y$,

$$f^{-1}\left({Int}_{\sigma }B\right)\subseteq {hInt}_X\left(f^{-1}\left({hCl}_Y\left({Int}_{\sigma }B\right)\right)\right).$$

Proof. 

Use the identifications

$$m_X={\tau }_h\left(X\right),{\sigma }_1=\sigma ,{\sigma }_2={\sigma }_h\left(Y\right).$$

Then $m_X$-interior and $m_X$-closure are precisely ${hInt}_X$ and ${hCl}_X$, while ${}_{2}Cl$ and ${}_{2}Int$ are ${hCl}_Y$ and ${hInt}_Y$. Hence Definition 3.5 of Noiri and Popa becomes exactly Definition 6. Under the same dictionary, [23, Theorem 3.4(5)] gives (ii), whereas [23, Theorem 3.1(2)] gives (iv); both are equivalent to (i) in that paper.

It remains only to note that (iii) is a reparametrization of (iv): (iv) implies (iii) by taking $B=F$ with F$\sigma$-closed, and (iii) implies (iv) by taking $F={Cl}_{\sigma }\left(B\right)$. Likewise, (v) is equivalent to (ii), since ${Int}_{\sigma }\left(B\right)$ is open and every open V satisfies $V={Int}_{\sigma }\left(V\right)$. □

A closure inclusion that one might be tempted to add to the theorem is not equivalent to weak h-continuity.

Example 3.

Let $X=\{a,b,c\}$ with

$$\tau =\{⌀,\{a\},\{a,b\},X\},$$

let $Y=\{0,1\}$ with $\sigma =\{⌀,\{0\},Y\}$, and put $f\left(a\right)=f\left(b\right)=0$, $f\left(c\right)=1$. Then

$${\tau }_h\left(X\right)=\{⌀,\left\{a\right\},\left\{b\right\},\{a,b\},\{b,c\},X\},{\sigma }_h\left(Y\right)=\mathcal{P}\left(Y\right).$$

The function is weakly h-continuous: at a and b the open set $\left\{0\right\}$ can be handled by the h-open neighbourhoods $\left\{a\right\}$ and $\left\{b\right\}$, respectively. Nevertheless, for $V=\left\{0\right\}$,

$${hCl}_X\left(f^{-1}\left(V\right)\right)={hCl}_X\left(\{a,b\}\right)=X\neg \subseteq \{a,b\}=f^{-1}\left({hCl}_Y\left(V\right)\right).$$

Thus the condition ${hCl}_X\left(f^{-1}\left(V\right)\right)\subseteq f^{-1}\left({hCl}_Y\left(V\right)\right)$ for open V is strictly stronger and must not be included among the equivalent characterizations.

Regular targets collapse weak continuity to ordinary continuity, both in the classical theory [13] and in the minimal-structure setting [23, Theorem 4.1]. In the present framework this yields the following useful consequence.

Proposition 4.

If $(Y,\sigma )$ is regular, then a function $f:(X,\tau )\to (Y,\sigma )$ is weakly h-continuous if and only if it is h-continuous.

Proof. 

By Proposition 3, weak h-continuity of f implies classical weak continuity of

$$f:(X,{\tau }_h\left(X\right))⟶(Y,\sigma ).$$

Since Y is regular, Levine’s theorem [13] makes this function continuous. By the definition of h-continuity, this is precisely the statement that f is h-continuous. Conversely, every h-continuous function is weakly h-continuous, since $V\subseteq {hCl}_Y\left(V\right)$ for every ordinary open set $V\in \sigma$. □

Proposition 5

(Composition principles). Let $f:(X,\tau )\to (Y,\sigma )$ be weakly h-continuous.

(i)

If $p:(W,\mu )\to (X,\tau )$ is h-irresolute, then $f\circ p$ is weakly h-continuous.

(ii)

If $g:(Y,\sigma )\to (Z,\rho )$ is both ordinarily continuous and h-irresolute, then $g\circ f$ is weakly h-continuous.

Proof. 

For (i), fix $z\in W$ and let $V\in \sigma$ contain $f\left(p\right(z\left)\right)$. By weak h-continuity of f, choose $U\in {\tau }_h\left(X\right)$ with $p\left(z\right)\in U$ and $f\left(U\right)\subseteq {hCl}_Y\left(V\right)$. Since p is h-irresolute, $p^{-1}\left(U\right)\in {\mu }_h\left(W\right)$ and $z\in p^{-1}\left(U\right)$. Moreover,

$$(f\circ p)\left(p^{-1}\left(U\right)\right)\subseteq f\left(U\right)\subseteq {hCl}_Y\left(V\right),$$

which is the required condition.

For (ii), fix $x\in X$ and let $O\in \rho$ contain $g\left(f\right(x\left)\right)$. Ordinary continuity of g makes $g^{-1}\left(O\right)$ an ordinary open neighbourhood of $f\left(x\right)$. Choose $U\in {\tau }_h\left(X\right)$ with $x\in U$ and

$$f\left(U\right)\subseteq {hCl}_Y\left(g^{-1}\left(O\right)\right).$$

Because g is h-irresolute, it is continuous between the associated spaces; hence

$$g\left({hCl}_Y\left(g^{-1}\left(O\right)\right)\right)\subseteq {hCl}_Z\left(g\left(g^{-1}\left(O\right)\right)\right)\subseteq {hCl}_Z\left(O\right).$$

Thus $(g\circ f)\left(U\right)\subseteq {hCl}_Z\left(O\right)$. □

Definition 7.

A space $(X,\tau )$ ish-connectedif it cannot be written as the disjoint union of two nonempty h-open sets. Equivalently, $(X,{\tau }_h\left(X\right))$ is connected.

Preservation of connectedness under weakly continuous surjections is classical [18], and analogous statements hold for weakly m-continuous functions [23,26]. The associated topology makes the corresponding h-statement immediate.

Theorem 3.

If $f:(X,\tau )\to (Y,\sigma )$ is weakly h-continuous and surjective and X is h-connected, then $(Y,\sigma )$ is connected.

Proof. 

By Definition 7, the associated space $(X,{\tau }_h\left(X\right))$ is connected. Proposition 3 shows that

$$f:(X,{\tau }_h\left(X\right))⟶(Y,\sigma )$$

is classically weakly continuous, and it is surjective by hypothesis. Noiri’s connectedness-preservation theorem [18] therefore implies that $(Y,\sigma )$ is connected. □

Definition 8.

A space $(Y,\sigma )$ isultra h-Urysohnif for each pair of distinct points $y_1,y_2\in Y$ there exist ordinary open sets $V_i\in \sigma$ with $y_i\in V_i$ and

$${hCl}_Y\left(V_1\right)\cap {hCl}_Y\left(V_2\right)=⌀.$$

Equivalently, distinct points admit ordinary open neighbourhoods whose closures in the associated topology $(Y,{\sigma }_h\left(Y\right))$ are disjoint.

The general minimal/bitopological theory already contains a closely related separation principle: pairwise weakly m-continuous injections into pairwise Urysohn spaces yield pairwise m-$T_2$ domains [23, Theorem 5.1]. A parallel result was obtained more recently in the ideal-topological $h\alpha$-setting by Almuhur, Al-Labadi and Bashir [2, Theorem 4.6]. The following h-open formulation uses only the one-sided mixed continuity required in Definition 6, so we keep the short direct proof.

Theorem 4.

Let $f:(X,\tau )\to (Y,\sigma )$ be injective and weakly h-continuous. If Y is ultra h-Urysohn, then X is h-$T_2$.

Proof. 

Let $x_1\ne x_2$. Choose ordinary open sets $V_i\ni f\left(x_i\right)$ such that ${hCl}_Y\left(V_1\right)\cap {hCl}_Y\left(V_2\right)=⌀$. Weak h-continuity gives $U_i\in {\tau }_h\left(X\right)$ with $x_i\in U_i$ and $f\left(U_i\right)\subseteq {hCl}_Y\left(V_i\right)$. If $z\in U_1\cap U_2$, then $f\left(z\right)$ belongs to both displayed h-closures, a contradiction. Thus $U_1\cap U_2=⌀$, proving that X is h-$T_2$ in the sense of [6,12]. □

Definition 9.

Let $A\subseteq X$, endowed with the original subspace topology. A weakly h-continuous function $r:X\to A$ is aweakly h-continuous retractionif $r\left(a\right)=a$ for every $a\in A$.

Theorem 5.

If $(X,\tau )$ is Hausdorff and $r:X\to A$ is a weakly h-continuous retraction, then A is h-closed in X (equivalently, closed in X).

Proof. 

By Proposition 2, ${\tau }_h\left(X\right)=\tau$; moreover the subspace A is Hausdorff, so its associated topology is its ordinary subspace topology. Thus all h-open sets and h-closures occurring in this proof reduce to the corresponding ordinary notions. Suppose $x\in {Cl}_{\tau }\left(A\right)\setminus A$. Then $r\left(x\right)\ne x$. Choose disjoint ordinary open sets $P\ni x$ and $V\ni r\left(x\right)$. Since $P\cap V$ is empty, $x\notin {Cl}_{\tau }\left(V\right)$, and

$$N:=X\setminus {Cl}_{\tau }\left(V\right)$$

is an open neighbourhood of x. The set $V\cap A$ is an open neighbourhood of $r\left(x\right)$ in A. Weak h-continuity yields an open neighbourhood H of x in X such that

$$r\left(H\right)\subseteq {Cl}_A(V\cap A)\subseteq A\cap {Cl}_X\left(V\right).$$

Since $x\in {Cl}_X\left(A\right)$, the open neighbourhood $N\cap H$ meets A; choose $a\in A\cap N\cap H$. Then $r\left(a\right)=a\in N$, whereas $a\in H$ gives $r\left(a\right)\in {Cl}_X\left(V\right)$, a contradiction. Thus A is closed, hence h-closed. □

4. Weakly H-Irresolute Functions

The fully associated analogue of weak continuity is useful both conceptually and technically.

Definition 10.

A function $f:(X,\tau )\to (Y,\sigma )$ isweakly h-irresoluteif, for every $x\in X$ and every $V\in {\sigma }_h\left(Y\right)$ with $f\left(x\right)\in V$, there exists $U\in {\tau }_h\left(X\right)$ with $x\in U$ such that

$$f\left(U\right)\subseteq {hCl}_Y\left(V\right).$$

Proposition 6.

A function $f:(X,\tau )\to (Y,\sigma )$ is weakly h-irresolute if and only if

$$f:(X,{\tau }_h\left(X\right))⟶(Y,{\sigma }_h\left(Y\right))$$

is weakly continuous in the classical sense. Moreover,

$$h-\mathrm{irresolute}⟹\mathrm{weakly}h-\mathrm{irresolute}⟹\mathrm{weakly}h-\mathrm{continuous}.$$

If Y is h-fixed, the last two weak notions coincide.

Proof. 

The first assertion is simply Definition 10 read in the two associated topologies. If f is h-irresolute and $V\in {\sigma }_h\left(Y\right)$, take $U=f^{-1}\left(V\right)$; then $f\left(U\right)\subseteq V\subseteq {hCl}_Y\left(V\right)$. The second implication follows from $\sigma \subseteq {\sigma }_h\left(Y\right)$. If ${\sigma }_h\left(Y\right)=\sigma$, the two weak definitions have the same neighbourhoods. □

The regular-target collapse for weak continuity has an immediate fully associated counterpart in the h-setting.

Proposition 7.

If the associated space $(Y,{\sigma }_h\left(Y\right))$ is regular, then weak h-irresoluteness and h-irresoluteness are equivalent.

Proof. 

By Proposition 6, weak h-irresoluteness is classical weak continuity of

$$f:(X,{\tau }_h\left(X\right))⟶(Y,{\sigma }_h\left(Y\right)).$$

Regularity of the target and Levine’s theorem [13] make this map continuous, which is exactly h-irresoluteness. The converse follows immediately from $V\subseteq {hCl}_Y\left(V\right)$ for every $V\in {\sigma }_h\left(Y\right)$. □

Proposition 8

(Composition). Let $f:(X,\tau )\to (Y,\sigma )$ and $g:(Y,\sigma )\to (Z,\rho )$.

(i)

If f is weakly h-irresolute and g is h-irresolute, then $g\circ f$ is weakly h-irresolute.

(ii)

If f is h-irresolute and g is weakly h-irresolute, then $g\circ f$ is weakly h-irresolute.

Proof. 

For (i), let $V\in {\rho }_h\left(Z\right)$ contain $g\left(f\right(x\left)\right)$. Since g is h-irresolute, $g^{-1}\left(V\right)\in {\sigma }_h\left(Y\right)$. Choose $U\in {\tau }_h\left(X\right)$ containing x with

$$f\left(U\right)\subseteq {hCl}_Y\left(g^{-1}\left(V\right)\right).$$

Continuity of $g:(Y,{\sigma }_h\left(Y\right))\to (Z,{\rho }_h\left(Z\right))$ gives

$$(g\circ f)\left(U\right)\subseteq g\left({hCl}_Y\left(g^{-1}\left(V\right)\right)\right)\subseteq {hCl}_Z\left(V\right).$$

For (ii), fix $x\in X$ and $V\in {\rho }_h\left(Z\right)$ with $g\left(f\right(x\left)\right)\in V$. Weak h-irresoluteness of g gives $W\in {\sigma }_h\left(Y\right)$ with $f\left(x\right)\in W$ and $g\left(W\right)\subseteq {hCl}_Z\left(V\right)$. Since f is h-irresolute, $f^{-1}\left(W\right)\in {\tau }_h\left(X\right)$ and contains x; moreover,

$$(g\circ f)\left(f^{-1}\left(W\right)\right)\subseteq g\left(W\right)\subseteq {hCl}_Z\left(V\right).$$

Thus $g\circ f$ is weakly h-irresolute. □

The associated-topology interpretation of h-connected and hyperconnected structures is developed further in [11].

Connectedness obeys the same preservation principle in the fully associated setting.

Theorem 6.

If $f:X\to Y$ is a surjective weakly h-irresolute function and X is h-connected, then Y is h-connected.

Proof. 

Proposition 6 identifies f with a surjective weakly continuous function

$$(X,{\tau }_h\left(X\right))⟶(Y,{\sigma }_h\left(Y\right)).$$

The domain is connected because X is h-connected. Applying [18] shows that $(Y,{\sigma }_h\left(Y\right))$ is connected, which is exactly h-connectedness of Y. □

5. Graphs and Product Compatibility

The graph of $f:X\to Y$ is denoted by

$$G\left(f\right)=\left\{\right(x,f\left(x\right)):x\in X\}.$$

One must distinguish the topology ${(\tau \times \sigma )}_h$ from ${\tau }_h\left(X\right)\times {\sigma }_h\left(Y\right)$; see [8,9]. Graph-map characterizations are a recurring feature of closure-based weak continuity; for example, Noiri proved the corresponding equivalence for $\theta$-precontinuous functions [21]. The next theorem is the h-specific version and requires explicit product compatibility.

Theorem 7.

Assume that $(X,\tau )$ and $(Y,\sigma )$ are h-product compatible. Let $g:X\to X\times Y$ be the graph map $g\left(x\right)=(x,f(x\left)\right)$. Then f is weakly h-continuous if and only if

$$g:(X,\tau )⟶(X\times Y,\tau \times \sigma )$$

is weakly h-continuous.

Proof. 

Suppose first that f is weakly h-continuous. Let $W\in \tau \times \sigma$ contain $g\left(x\right)$. Choose $O\in \tau$ and $V\in \sigma$ with

$$(x,f(x\left)\right)\in O\times V\subseteq W.$$

There exists $H\in {\tau }_h\left(X\right)$ with $x\in H$ and $f\left(H\right)\subseteq {hCl}_Y\left(V\right)$. Put $U=O\cap H\in {\tau }_h\left(X\right)$. Product compatibility gives

$${hCl}_{X\times Y}(O\times V)={hCl}_X\left(O\right)\times {hCl}_Y\left(V\right).$$

Hence

$$g\left(U\right)\subseteq O\times {hCl}_Y\left(V\right)\subseteq {hCl}_{X\times Y}(O\times V)\subseteq {hCl}_{X\times Y}\left(W\right),$$

so g is weakly h-continuous.

Conversely, let $V\in \sigma$ contain $f\left(x\right)$. Then $X\times V$ is an ordinary open neighbourhood of $g\left(x\right)$. Weak h-continuity of g gives $U\in {\tau }_h\left(X\right)$, $x\in U$, such that

$$g\left(U\right)\subseteq {hCl}_{X\times Y}(X\times V).$$

By compatibility,

$${hCl}_{X\times Y}(X\times V)=X\times {hCl}_Y\left(V\right),$$

and therefore $f\left(U\right)\subseteq {hCl}_Y\left(V\right)$. □

Closed-graph theorems for weakly continuous functions into Hausdorff spaces [18,19] provide the correct model here. In the h-setting, one must first work in the appropriate associated product; an h-product compatibility hypothesis is needed only to transfer that closedness back to the original product.

Theorem 8.

If $f:(X,\tau )\to (Y,\sigma )$ is weakly h-continuous and Y is Hausdorff, then $G\left(f\right)$ is closed in

$$(X,{\tau }_h\left(X\right))\times (Y,\sigma ).$$

If, in addition, $(X,\tau )$ and $(Y,\sigma )$ are h-product compatible, then $G\left(f\right)$ is h-closed in the original product $X\times Y$.

Proof. 

By Proposition 3, $f:(X,{\tau }_h\left(X\right))\to (Y,\sigma )$ is classically weakly continuous. Since Y is Hausdorff, Noiri’s closed-graph theorem [19] yields the first assertion.

For the second assertion, Hausdorff rigidity gives ${\sigma }_h\left(Y\right)=\sigma$. Product compatibility therefore identifies

$${(\tau \times \sigma )}_h(X\times Y)={\tau }_h\left(X\right)\times \sigma .$$

Hence a graph closed in the product appearing in the first assertion is closed in the associated topology of the original product, i.e. it is h-closed. □

For the stronger point-separation results it is convenient to use the following graph condition. Strongly m-closed graphs were studied together with weakly m-continuous functions by Popa and Noiri [26]; the condition below is tailored to the mixed h-setting. The codomain neighbourhood is deliberately required to be ordinary open, so that it is compatible with Definition 6.

Definition 11.

The graph $G\left(f\right)$ isultra h-closedif, for every $(x,y)\notin G\left(f\right)$, there exist $U\in {\tau }_h\left(X\right)$ with $x\in U$ and an ordinary open set $V\in \sigma$ with $y\in V$ such that

$$(U\times {hCl}_Y\left(V\right))\cap G\left(f\right)=⌀.$$

Equivalently,

$$f\left(U\right)\cap {hCl}_Y\left(V\right)=⌀.$$

Proposition 9.

If $f:X\to Y$ is weakly h-continuous and Y is ultra h-Urysohn, then $G\left(f\right)$ is ultra h-closed.

Proof. 

Let $(x,y)\notin G\left(f\right)$. Choose ordinary open sets $V\ni f\left(x\right)$ and $W\ni y$ with ${hCl}_Y\left(V\right)\cap {hCl}_Y\left(W\right)=⌀$. Weak h-continuity gives $U\in {\tau }_h\left(X\right)$ with $x\in U$ and $f\left(U\right)\subseteq {hCl}_Y\left(V\right)$. Hence $f\left(U\right)\cap {hCl}_Y\left(W\right)=⌀$. □

Proposition 10.

If $f:X\to Y$ is injective and weakly h-continuous and $G\left(f\right)$ is ultra h-closed, then X is h-$T_2$.

Proof. 

Let $x\ne y$. Since f is injective, $(x,f(y\left)\right)\notin G\left(f\right)$. Choose $U\in {\tau }_h\left(X\right)$ with $x\in U$ and $V\in \sigma$ with $f\left(y\right)\in V$ such that

$$f\left(U\right)\cap {hCl}_Y\left(V\right)=⌀.$$

Weak h-continuity at y gives $H\in {\tau }_h\left(X\right)$ with $y\in H$ and $f\left(H\right)\subseteq {hCl}_Y\left(V\right)$. If $z\in U\cap H$, then $f\left(z\right)\in f\left(U\right)\cap {hCl}_Y\left(V\right)$, a contradiction. Thus $U\cap H=⌀$. □

Proposition 11.

If $f:X\to Y$ is h-continuous and Y is Hausdorff, then $G\left(f\right)$ is ultra h-closed.

Proof. 

Let $(x,y)\notin G\left(f\right)$. Choose disjoint ordinary open sets $O\ni f\left(x\right)$ and $V\ni y$. Since the open neighbourhood O of $f\left(x\right)$ is disjoint from V, we have $f\left(x\right)\notin {Cl}_{\sigma }\left(V\right)$. Hence $W:=Y\setminus {Cl}_{\sigma }\left(V\right)$ is an ordinary open neighbourhood of $f\left(x\right)$. By h-continuity,

$$U:=f^{-1}\left(W\right)\in {\tau }_h\left(X\right),x\in U.$$

Since Y is Hausdorff, ${hCl}_Y\left(V\right)={Cl}_{\sigma }\left(V\right)$ by Proposition 2. Therefore $f\left(U\right)\cap {hCl}_Y\left(V\right)=⌀$. □

The fully associated closed-graph statement requires Hausdorffness only of the associated codomain. Product compatibility enters, as before, only when the conclusion is transferred to the original product.

Theorem 9.

Let $f:X\to Y$ be weakly h-irresolute. If Y is h-$T_2$, then $G\left(f\right)$ is closed in

$$(X,{\tau }_h\left(X\right))\times (Y,{\sigma }_h\left(Y\right)).$$

If the pair is h-product compatible, then $G\left(f\right)$ is h-closed in the original product.

Proof. 

By Proposition 6, $f:(X,{\tau }_h\left(X\right))\to (Y,{\sigma }_h\left(Y\right))$ is classically weakly continuous. The h-$T_2$ hypothesis means precisely that $(Y,{\sigma }_h\left(Y\right))$ is Hausdorff, so [19] gives the first assertion. If the pair is h-product compatible, then

$${(\tau \times \sigma )}_h(X\times Y)={\tau }_h\left(X\right)\times {\sigma }_h\left(Y\right),$$

and closedness in the displayed product is exactly h-closedness in the original product. □

6. Relative H-Compactness and H-$\theta$ Image Properties

The associated topology also clarifies the correct meaning of compactness for a subset of an ambient space.

Definition 12.

A subset $A\subseteq X$ ish-compact relative to Xif every cover of A by members of ${\tau }_h\left(X\right)$ has a finite subcover. Equivalently, $(A,{\tau }_h\left(X\right){|}_A)$ is compact.

For $A=X$ this is the usual h-compactness of [28]. If A is h-subspace compatible, relative h-compactness is equivalent to intrinsic h-compactness of the original subspace $(A,\tau {|}_A)$; without compatibility the two notions need not coincide, as emphasized in [8,11].

We use two cover notions in the codomain. The terminology is chosen to avoid confusion with the point-set notion of an h-$\theta$-closed set from Definition 3.

Definition 13.

A subset $B\subseteq Y$ is

(i)

h-$\theta$-cover-closed relative to Yif every cover ${\left\{V_{\alpha }\right\}}_{\alpha \in \Lambda }$ of B by members of ${\sigma }_h\left(Y\right)$ has a finite subfamily such that

$$B\subseteq {\displaystyle \bigcup _{j=1}^n}{hCl}_Y\left(V_{{\alpha }_j}\right);$$

(ii)

weakly h-$\theta$-cover-closed relative to Yif the same condition is required only for covers by ordinary σ-open sets.

Finite-cover preservation under weak generalized continuity has been studied in the minimal-structure setting [23,26]; a related pattern for $\theta$-precontinuous functions appears in [21, Theorem 5.3]. The following result is the corresponding formulation for weak h-continuity.

Theorem 10.

Let $f:X\to Y$ be weakly h-continuous and let $A\subseteq X$ be h-compact relative to X. Then $f\left(A\right)$ is weakly h-θ-cover-closed relative to Y.

Proof. 

This is the specialization of [23] obtained by taking

$$m_X={\tau }_h\left(X\right),{\sigma }_1=\sigma ,{\sigma }_2={\sigma }_h\left(Y\right).$$

With these choices, $m_X$-compactness of A is exactly relative h-compactness, and $(1,2)$-quasi H-closedness of $f\left(A\right)$ means precisely that every ordinary open cover of $f\left(A\right)$ has a finite subfamily whose ${\sigma }_h\left(Y\right)$-closures cover $f\left(A\right)$. This is Definition 13(ii). □

Corollary 1.

If $f:X\to Y$ is a surjective weakly h-continuous function and X is h-compact, then Y is weakly h-θ-cover-closed.

Proof. 

Apply Theorem 10 with $A=X$ and use $f\left(X\right)=Y$. □

The same finite-cover principle also has a fully associated version.

Theorem 11.

Let $f:X\to Y$ be weakly h-irresolute and let $A\subseteq X$ be h-compact relative to X. Then $f\left(A\right)$ is h-θ-cover-closed relative to Y.

Proof. 

Apply [23, Theorem 5.3] with

$$m_X={\tau }_h\left(X\right),{\sigma }_1={\sigma }_2={\sigma }_h\left(Y\right).$$

Then $(1,2)$-weak m-continuity is exactly weak h-irresoluteness, while the resulting quasi H-closedness condition is precisely h-$\theta$-cover-closedness from Definition 13(i). □

Corollary 2.

If $f:X\to Y$ is surjective and weakly h-irresolute and X is h-compact, then Y is h-θ-cover-closed.

Proof. 

Apply Theorem 11 with $A=X$ and use $f\left(X\right)=Y$. □

The ultra graph condition yields a point-set h-$\theta$-closed image, which is stronger in a different direction.

Theorem 12.

Assume that $G\left(f\right)$ is ultra h-closed and that $A\subseteq X$ is h-compact relative to X. Then $f\left(A\right)$ is h-θ-closed in Y.

Proof. 

Let $y\notin f\left(A\right)$. For each $x\in A$, Definition 11 gives $U_x\in {\tau }_h\left(X\right)$ containing x and an ordinary open $V_x\in \sigma$ containing y such that

$$f\left(U_x\right)\cap {hCl}_Y\left(V_x\right)=⌀.$$

Choose $x_1,\dots ,x_n\in A$ with $A\subseteq U_{x_1}\cup \dots \cup U_{x_n}$ and put

$$V:=V_{x_1}\cap \dots \cap V_{x_n}.$$

Then V is an ordinary, hence h-open, neighbourhood of y. By monotonicity of h-closure, ${hCl}_Y\left(V\right)\subseteq {hCl}_Y\left(V_{x_j}\right)$ for each j, and therefore

$$f\left(A\right)\cap {hCl}_Y\left(V\right)=⌀.$$

Thus $y\notin {hCl}_{\theta }\left(f\left(A\right)\right)$. Hence ${hCl}_{\theta }\left(f\left(A\right)\right)\subseteq f\left(A\right)$. The reverse inclusion is automatic: if $y\in f\left(A\right)$ and $V\in {\sigma }_h\left(Y\right)$ contains y, then $y\in V\subseteq {hCl}_Y\left(V\right)$, so ${hCl}_Y\left(V\right)\cap f\left(A\right)\ne ⌀$. Therefore $f\left(A\right)\subseteq {hCl}_{\theta }\left(f\left(A\right)\right)$, and equality follows. □

7. The Set of Points of Failure

Definition 14.

For $A\subseteq X$, theh-frontierof A is

$${hFr}_X\left(A\right):={hCl}_X\left(A\right)\cap {hCl}_X(X\setminus A),$$

that is, the ordinary frontier of A in the associated space $(X,{\tau }_h\left(X\right))$.

Let $D_{wh}\left(f\right)$ denote the set of points at which f is not weakly h-continuous. Its description in terms of h-frontiers is the natural counterpart of the failure-set characterization for weakly m-continuous functions due to Noiri and Popa [23, Theorem 5.5].

Theorem 13.

For every function $f:(X,\tau )\to (Y,\sigma )$,

$$D_{wh}\left(f\right)=\bigcup _{V\in \sigma }\left[f^{-1}\left(V\right)\cap {hFr}_X\left(f^{-1}\left({hCl}_Y\left(V\right)\right)\right)\right].$$

Equivalently, $x\in D_{wh}\left(f\right)$ if and only if there exists $V\in \sigma$ with $f\left(x\right)\in V$ such that

$$x\in {hFr}_X\left(f^{-1}\left({hCl}_Y\left(V\right)\right)\right).$$

Proof. 

Apply [23, Theorem 5.5] with the same dictionary used in Theorem 2:

$$m_X={\tau }_h\left(X\right),{\sigma }_1=\sigma ,{\sigma }_2={\sigma }_h\left(Y\right).$$

Then the $m_X$-frontier is ${hFr}_X$, ${}_{2}Cl\left(V\right)={hCl}_Y\left(V\right)$, and $(1,2)$-weak m-continuity is weak h-continuity. The cited theorem therefore gives the pointwise criterion in the second display. Since $f\left(x\right)\in V$ is equivalent to $x\in f^{-1}\left(V\right)$, taking the union over $V\in \sigma$ yields the first formula. □

8. Concluding Remarks

The associated topology gives a clean framework for weak h-continuity, but the main lesson is that two levels must be kept separate. Weak h-continuity is a mixed condition: the domain uses ${\tau }_h\left(X\right)$ while the codomain neighbourhood is ordinary open and its closure is taken in ${\sigma }_h\left(Y\right)$. Weak h-irresoluteness is the fully associated version and is therefore exactly classical weak continuity between associated spaces. This distinction explains both the valid transfer results and the places where additional hypotheses are necessary.

The finite examples show that weak h-continuity is genuinely weaker than h-continuity and that classical weak continuity from the associated domain is not sufficient in a non-h-fixed codomain. At the same time, the identification with $(1,2)$-weak m-continuity makes clear which parts of the theory are specializations of the general minimal/bitopological results of Popa and Noiri [23,26]. In particular, the basic equivalent characterizations, the compact-image cover theorem and the frontier description belong to that general framework. The genuinely h-specific issues arise from the associated topology and from comparing it with the original topology: product and subspace compatibility, the mixed-versus-fully-associated distinction, the finite counterexamples, and the resulting graph and separation statements. These points place the paper coherently within both the classical weak-continuity literature and the recent h-open-set programme.