On h-D₁ Topological Spaces

1. Introduction

The systematic study of generalized open sets began with notions such as semi-open, $\alpha$-open, preopen and b-open sets [3,6,7,8]. These classes have generated numerous weakened forms of continuity, compactness and separation and provide the broader setting for the present work. Abbas introduced the notion of an h-open set, together with the associated notions of h-closure, h-continuity and h-irresoluteness [1]. Early corrections to some statements and examples were pointed out by Cakalli and Ince Dagci [4]. The theory was subsequently revisited in a systematic corrigendum and addendum by Sharma, Saproo, Billawria and Digra [10]. That work rectified an important characterization from the initial paper and established, in particular, that the family of all h-open subsets of a topological space $(X,\tau )$ forms a topology

$${\tau }_h=\{A\subseteq X:A\mathrm{is}h-\mathrm{open}\},$$

without requiring a $T_{1/2}$ hypothesis.

This fact changes the natural viewpoint. Rather than treating h-open sets merely as generalized open sets, one may regard them as the open sets of the associated space $(X,{\tau }_h)$. Consequently, an h-topological property is often a classical topological property computed with respect to ${\tau }_h$. The comparison between $\tau$ and ${\tau }_h$, however, remains meaningful because the two topologies may differ substantially. The role of weak separation assumptions in this context is related to the classical $T_{1/2}$ condition of Dunham [5], although the topological character of ${\tau }_h$ does not depend on that hypothesis [10].

The literature has also begun to apply h-open sets beyond elementary point-set topology. Examples include $h\alpha$-open sets and their associated mappings [2], h-irresolute topological vector spaces [11], and h-topological groups [9]. These developments reinforce the usefulness of a precise separation theory for the associated topology.

Tong introduced the separation axiom $D_1$, lying between $T_0$ and $T_1$, by using differences of open sets [12]. The aim of the present paper is to develop the corresponding theory based on h-open sets. We introduce h-difference sets and the axioms h-$D_1$ and h-$D_2$, prove that the latter two conditions are equivalent, and study their relationships with h-$T_i$ and h-symmetric spaces. We also investigate preservation under natural classes of mappings and provide finite examples illustrating the distinction between the original topology and the associated h-topology.

Section 2 recalls the necessary terminology. Section 3 develops the theory of h-difference sets and h-$D_i$ spaces, including their relation with h-$T_i$, h-$R_0$ and h-symmetry. Section 4 records preservation and product results. A final section summarizes the role of the associated topology ${\tau }_h$.

2. Preliminaries

Throughout, $(X,\tau )$ and $(Y,\sigma )$ denote topological spaces, with no separation axiom assumed unless explicitly stated. The ordinary interior and closure with respect to the original topology are denoted by Int and Cl, respectively.

Definition 1. 

A subset $A\subseteq X$ is calledh-open[1] if

$$A\subseteq Int(A\cup U)$$

for every nonempty proper open set $U\in \tau$. The complement of an h-open set is calledh-closed. We write $h\mathcal{O}\left(X\right)$ and $h\mathcal{C}\left(X\right)$ for the families of all h-open and h-closed subsets of X, respectively.

The definition above is the journal formulation used in [1,10]; the corrections in [4,10] should be kept in mind whenever results from the earliest version of the theory are invoked.

By [10], $h\mathcal{O}\left(X\right)$ is a topology on X; throughout the paper it is denoted by ${\tau }_h$. Thus

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

Definition 2. 

A point $x\in X$ is anh-cluster point of $A\subseteq X$ if every h-open set containing x meets A. The set of all h-cluster points of A is theh-closure of A, denoted by $hCl\left(A\right)$. Equivalently, $hCl\left(A\right)$ is the closure of A in $(X,{\tau }_h)$.

A subset $N\subseteq X$ is an h-neighborhood of x if there exists $V\in {\tau }_h$ such that $x\in V\subseteq N$.

Lemma 1. 

For all $A,B\subseteq X$ and every family $\{A_i:i\in I\}$ of subsets of X, the following hold:

(1)

arbitrary intersections of h-closed sets are h-closed;

(2)

arbitrary unions of h-open sets are h-open;

(3)

A is h-closed if and only if $A=hCl\left(A\right)$;

(4)

$hCl\left(A\right)=\bigcap \{F\in h\mathcal{C}(X):A\subseteq F\}$;

(5)

$A\subseteq hCl\left(A\right)$;

(6)

if $A\subseteq B$, then $hCl\left(A\right)\subseteq hCl\left(B\right)$;

(7)

$hCl\left(A\right)$ is h-closed.

Proof. 

These are the standard closure properties in the topological space $(X,{\tau }_h)$. □

Definition 3. 

For $i\in \{0,1,2\}$, the space $(X,\tau )$ is called h-$T_i$ if $(X,{\tau }_h)$ is $T_i$. Explicitly:

(i)

h-$T_0$: distinct points are distinguished by an h-open set;

(ii)

h-$T_1$: for distinct $x,y$, each point has an h-open neighborhood missing the other;

(iii)

h-$T_2$: distinct points have disjoint h-open neighborhoods.

Definition 4. 

A mapping $f:(X,\tau )\to (Y,\sigma )$ ish-irresolute if $f^{-1}\left(V\right)\in {\tau }_h\left(X\right)$ for every $V\in {\tau }_h\left(Y\right)$. It ish-continuousif $f^{-1}\left(V\right)\in {\tau }_h\left(X\right)$ for every $V\in \sigma$.

Thus an h-irresolute mapping is precisely a continuous mapping from $(X,{\tau }_h\left(X\right))$ to $(Y,{\tau }_h\left(Y\right))$. This interpretation is also the natural framework for the algebraic applications developed in [9,11].

3. h-Difference Sets and h-$D_i$ Spaces

Definition 5. 

A subset $G\subseteq X$ is anh-difference set, briefly an $hD$-set, if

$$G=U\setminus V$$

for some $U,V\in {\tau }_h$ with $U\ne X$.

Every proper h-open set is an $hD$-set, by taking $V=⌀$. The converse fails in general.

Definition 6. 

A space $(X,\tau )$ is called h-$D_1$ if for every pair of distinct points $x,y\in X$ there are $hD$-sets G and E such that

$$x\in G,y\notin G,y\in E,x\notin E.$$

It is called h-$D_2$ if the sets G and E can be chosen disjoint.

Remark 1. 

Every h-$T_1$ space is h-$D_1$, every h-$T_2$ space is h-$D_2$, and every h-$D_2$ space is h-$D_1$.

3.1. Examples and Finite Computations

For finite spaces the associated topology can be computed directly from the definition. If $(X,\tau )$ is finite, then a subset $A\subseteq X$ belongs to ${\tau }_h$ if and only if

$$A\subseteq {Int}_{\tau }(A\cup U)\mathrm{for}\mathrm{every}U\in \tau \setminus \{⌀,X\}.$$

Thus one may test each subset of X against the finitely many nonempty proper open sets. Once ${\tau }_h$ has been determined, h-closures, $hD$-sets and the corresponding h-separation properties are obtained by applying the ordinary topological operations in $(X,{\tau }_h)$. The following examples illustrate both the computation and the extent to which passage from $\tau$ to ${\tau }_h$ may improve separation.

Example 1. 

Let

$$X=\{a,b,c\},\tau =\{⌀,X,\{a,b\left\}\right\}.$$

The only nonempty proper member of τ is $U=\{a,b\}$. For every $A\subseteq X$ one has

$$A\subseteq Int(A\cup U).$$

Indeed, if $c\notin A$, then $A\subseteq U=Int(A\cup U)$; if $c\in A$, then $A\cup U=X$ and $Int(A\cup U)=X$. Hence every subset of X is h-open and

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

Thus $(X,\tau )$ is h-$T_2$, and therefore h-$D_2$ and h-$D_1$. On the other hand, the original space $(X,\tau )$ is neither $T_0$ nor $T_1$ (and hence not $T_2$), since a and b have exactly the same open neighborhoods.

Remark 2. 

Example 1 shows that h-separation axioms may be strictly weaker than their counterparts in the original topology.

Example 2 

(An infinite example). Let A be an infinite set and choose $p\notin A$. Set

$$X=A\cup \left\{p\right\},\tau =\{⌀,A,X\}.$$

The only nonempty proper open set is A. If $B\subseteq A$, then $B\subseteq A=Int(B\cup A)$; if $p\in B$, then $B\cup A=X$ and hence $B\subseteq Int(B\cup A)=X$. Consequently every subset of X is h-open and ${\tau }_h=\mathcal{P}\left(X\right)$. Thus $(X,\tau )$ is h-$T_2$ and h-$D_1$, although the original topology is not even $T_0$ whenever A contains at least two points.

Remark 3. 

Examples 1 and 2 show that the passage from τ to ${\tau }_h$ may dramatically increase separation, even producing a discrete associated topology from a highly non-separated original space.

Theorem 1. 

A topological space is h-$D_1$ if and only if it is h-$D_2$.

Proof. 

Only the forward implication requires proof. Let $x\ne y$. Since X is h-$D_1$, there are

$$G=U\setminus V,E=W\setminus N,$$

where $U,V,W,N\in {\tau }_h$, $U\ne X$, $W\ne X$, $x\in G\setminus E$ and $y\in E\setminus G$.

If $x\notin W$, then either $y\notin U$ or $y\in V$. If $y\notin U$, set

$$G_x=U\setminus (V\cup W),G_y=W\setminus (N\cup U).$$

These are disjoint $hD$-sets containing x and y, respectively. If $y\in V$, then $G_x=U\setminus V$ and $G_y=V$ are disjoint $hD$-sets containing x and y. Notice that $V\ne X$, because $x\in U\setminus V$.

If $x\in W$, then $x\notin E=W\setminus N$ implies $x\in N$. Since $y\in W\setminus N$, the sets $W\setminus N$ and N are disjoint $hD$-sets containing y and x, respectively; moreover $N\ne X$ because $y\notin N$. Thus X is h-$D_2$. □

Proposition 1. 

Every h-$D_1$ space is h-$T_0$.

Proof. 

Let $x\ne y$. Choose an $hD$-set $G=U\setminus V$ such that $x\in G$ and $y\notin G$, where $U,V\in {\tau }_h$ and $U\ne X$. If $y\notin U$, then the h-open set U contains x and misses y. If $y\in U$, the condition $y\notin U\setminus V$ yields $y\in V$, while $x\notin V$; hence the h-open set V contains y and misses x. Therefore $(X,{\tau }_h)$ is $T_0$. □

Combining Remark 1, Theorem 1 and Proposition 1, we obtain the implication diagram in Figure 1. The double arrow records the equivalence between h-$D_1$ and h-$D_2$.

Proposition 2. 

The following are equivalent:

(i)

$(X,\tau )$ is h-$D_1$;

(ii)

$(X,{\tau }_h)$ is a classical $D_1$ space;

(iii)

$(X,\tau )$ is h-$D_2$;

(iv)

$(X,{\tau }_h)$ is a classical $D_2$ space.

Proof. 

The notions of $hD$-set and classical difference set in $(X,{\tau }_h)$ coincide by definition. The equivalences now follow from Theorem 1 and the corresponding classical terminology. □

3.2. The Associated-Topology Viewpoint

The construction used throughout the paper can be summarized by the chain

$$(X,\tau )⟼{\tau }_h⟼h-\mathrm{neighborhoods}\mathrm{and}h-\mathrm{closure}⟼hD-\mathrm{sets}⟼h-D_i\mathrm{axioms}.$$

Figure 2 records this dependence. It emphasizes that the original topology enters through the definition of the family of h-open sets, whereas all subsequent notions are ordinary topological notions in $(X,{\tau }_h)$.

Remark 4. 

Proposition 2 shows that the intrinsic h-$D_i$ theory is formally the classical $D_i$ theory of $(X,{\tau }_h)$. The genuinely new information therefore lies in the comparison of ${\tau }_h$ with the original topology τ and in conditions ensuring that properties pass from one topology to the other.

3.3. h-Symmetry

Definition 7. 

The space $(X,\tau )$ ish-symmetric if

$$x\in hCl\left(\right\{y\left\}\right)⟹y\in hCl\left(\right\{x\left\}\right)$$

for all $x,y\in X$.

A subset $A\subseteq X$ is called h-generalized closed, briefly $hg$-closed, if

$$hCl\left(A\right)\subseteq U$$

whenever $A\subseteq U$ and $U\in {\tau }_h$.

Theorem 2. 

A space $(X,\tau )$ is h-symmetric if and only if every singleton is $hg$-closed.

Proof. 

Assume first that X is h-symmetric. Let $\left\{x\right\}\subseteq U\in {\tau }_h$ and take $y\in hCl\left(\right\{x\left\}\right)$. By h-symmetry, $x\in hCl\left(\right\{y\left\}\right)$. If $y\notin U$, then the h-closed set $X\setminus U$ contains y, and therefore contains $hCl\left(\right\{y\left\}\right)$, contradicting $x\in U$. Hence $y\in U$ and $hCl\left(\right\{x\left\}\right)\subseteq U$.

Conversely, assume that every singleton is $hg$-closed and let $x\in hCl\left(\right\{y\left\}\right)$. If $y\notin hCl\left(\right\{x\left\}\right)$, then

$$U=X\setminus hCl\left(\right\{x\left\}\right)$$

is an h-open set containing y. Since $\left\{y\right\}$ is $hg$-closed, $hCl\left(\right\{y\left\}\right)\subseteq U$, which contradicts $x\in hCl\left(\right\{y\left\}\right)$. Thus $y\in hCl\left(\right\{x\left\}\right)$. □

Corollary 1. 

Every h-$T_1$ space is h-symmetric.

Proof. 

In an h-$T_1$ space each singleton is h-closed, hence $hg$-closed. Apply Theorem 2. □

Theorem 3. 

A space $(X,\tau )$ is h-$T_1$ if and only if it is both h-$T_0$ and h-symmetric.

Proof. 

The forward implication follows from Corollary 1. Conversely, let $x\ne y$. Since X is h-$T_0$, one of the points, say x, has an h-open neighborhood missing y. Equivalently, $x\notin hCl\left(\right\{y\left\}\right)$. By h-symmetry, $y\notin hCl\left(\right\{x\left\}\right)$. Hence there is also an h-open neighborhood of y missing x. Therefore X is h-$T_1$. □

Remark 5. 

Theorem 3 corrects the overly strong statement that h-symmetry alone implies h-$T_1$. As in ordinary topology, symmetry corresponds to an $R_0$-type condition; a $T_0$ hypothesis is needed to obtain $T_1$.

Definition 8. 

The space $(X,\tau )$ is called h-$R_0$ if, whenever x belongs to an h-open set U, one has $hCl\left(\right\{x\left\}\right)\subseteq U$. Equivalently, $(X,{\tau }_h)$ is an $R_0$ space.

Proposition 3. 

A space is h-$R_0$ if and only if it is h-symmetric.

Proof. 

Suppose first that X is h-$R_0$ and that $x\in hCl\left(\right\{y\left\}\right)$. If $y\notin hCl\left(\right\{x\left\}\right)$, then the h-open set $U=X\setminus hCl\left(\right\{x\left\}\right)$ contains y. By the h-$R_0$ property, $hCl\left(\right\{y\left\}\right)\subseteq U$, contradicting $x\in hCl\left(\right\{y\left\}\right)$. Thus $y\in hCl\left(\right\{x\left\}\right)$.

Conversely, suppose that X is h-symmetric, let $x\in U\in {\tau }_h$, and take $y\in hCl\left(\right\{x\left\}\right)$. By symmetry, $x\in hCl\left(\right\{y\left\}\right)$. If $y\notin U$, then the h-closed set $X\setminus U$ contains y and therefore contains $hCl\left(\right\{y\left\}\right)$, contradicting $x\in U$. Hence $y\in U$, and $hCl\left(\right\{x\left\}\right)\subseteq U$. □

Corollary 2. 

A space is h-$T_1$ if and only if it is both h-$T_0$ and h-$R_0$.

Proof. 

Combine Theorem 3 with Proposition 3. □

The relationships proved in this subsection are summarized in Figure 3.

3.4. The h-$R_H$ and h-$R_D$ Conditions

We next record two singleton-closure conditions that are naturally expressed in the associated topology. The notation $R_H$ is used here for the semisimple condition adopted in the companion discussion on h-regularity.

Definition 9. 

A space $(X,\tau )$ is calledh-$R_H$(orh-semisimple) if $hCl\left(\right\{x\left\}\right)$ is h-open for every $x\in X$. Equivalently, every closure of a singleton in $(X,{\tau }_h)$ is open in $(X,{\tau }_h)$.

Proposition 4. 

Every h-$R_H$ space is h-$R_0$.

Proof. 

Let $x\in U\in {\tau }_h$. Since $hCl\left(\right\{x\left\}\right)$ is the smallest h-closed set containing x, the h-$R_0$ condition is equivalent to the fact that every h-open neighborhood of x contains $hCl\left(\right\{x\left\}\right)$. In an h-$R_H$ space the specialization classes $hCl\left(\right\{x\left\}\right)$ are h-open as well as h-closed; hence each h-open set is saturated with respect to these classes, and therefore $hCl\left(\right\{x\left\}\right)\subseteq U$. □

Definition 10. 

A space $(X,\tau )$ is calledh-$R_D$if

$$hCl\left(\right\{x\left\}\right)\setminus \left\{x\right\}$$

is h-closed for every $x\in X$.

Example 3. 

Let

$$X=\{a,b,c\},\tau =\{⌀,X,\{b\left\}\right\}.$$

A direct computation gives

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

The associated topology is a partition topology, whose specialization classes are $\left\{b\right\}$ and $\{a,c\}$. Consequently the space is h-$R_0$. Moreover,

$$hCl\left(\right\{b\left\}\right)=\left\{b\right\},hCl\left(\right\{a\left\}\right)=hCl\left(\right\{c\left\}\right)=\{a,c\},$$

and all these sets belong to ${\tau }_h$. Hence the space is h-$R_H$.

On the other hand,

$$hCl\left(\right\{a\left\}\right)\setminus \left\{a\right\}=\left\{c\right\}.$$

The set $\left\{c\right\}$ is not h-closed, since its complement $\{a,b\}$ does not belong to ${\tau }_h$. Thus the space is not h-$R_D$. In particular,

$$h-R_H\neg ⟹h-R_D.$$

The same example is not h-$T_0$, because a and c have exactly the same h-open neighborhoods; hence it is not h-$T_1$ either.

Remark 6. 

The preceding example shows that the h-$R_H$ condition refines the singleton-closure structure of an h-$R_0$ space without forcing point separation. Thus h-$R_H$ should not be placed in a chain ending in h-$T_1$ unless an additional h-$T_0$ hypothesis is imposed. With that hypothesis, Corollary 2 yields h-$T_1$.

Figure 4. Relations among h-$R_H$, h-$R_0$, h-symmetry and h-$T_1$. The dashed implication requires the additional h-$T_0$ hypothesis.

Figure 4. Relations among h-$R_H$, h-$R_0$, h-symmetry and h-$T_1$. The dashed implication requires the additional h-$T_0$ hypothesis.

3.5. h-Compactness and the Associated Topology

Recall that a space is called h-compact if every cover by h-open sets has a finite subcover [10]. This is the direct h-analogue of ordinary compactness as treated in standard topology texts such as [13]. Since the h-open sets are exactly the members of ${\tau }_h$, this notion admits an immediate reformulation.

Proposition 5. 

A topological space $(X,\tau )$ is h-compact if and only if the associated space $(X,{\tau }_h)$ is compact.

Proof. 

The two definitions quantify over exactly the same families of sets, namely the open covers of $(X,{\tau }_h)$. □

Proposition 6. 

In an h-$T_2$ space, every h-compact subset is h-closed.

Proof. 

If $(X,\tau )$ is h-$T_2$, then $(X,{\tau }_h)$ is Hausdorff. By Proposition 5, an h-compact subset is compact in $(X,{\tau }_h)$. Compact subsets of Hausdorff spaces are closed, so the subset is closed in ${\tau }_h$, that is, h-closed. □

Remark 7. 

Proposition 6 identifies the precise classical mechanism behind the result. The conclusion is not automatic in an h-$D_1$ space, because $D_1$ is strictly weaker than Hausdorff separation. This distinction delineates the range in which familiar compactness theorems remain valid after passing to the associated topology.

Proposition 7. 

Let $(X,\tau )$ be an h-$D_1$ space, and let $Y\subseteq X$. If

$${\tau }_h\left(Y\right)={\tau }_h{\left(X\right)|}_Y,$$

then the subspace $(Y,\tau {|}_Y)$ is h-$D_1$.

Proof. 

Since X is h-$D_1$, the associated space $(X,{\tau }_h\left(X\right))$ is a classical $D_1$ space. The $D_1$ axiom is hereditary: if $x\ne y$ belong to Y, difference sets of X separating them restrict to difference sets of the subspace $(Y,{\tau }_h\left(X\right){|}_Y)$. By the compatibility hypothesis, this subspace topology is exactly ${\tau }_h\left(Y\right)$. Hence $(Y,{\tau }_h\left(Y\right))$ is $D_1$, which means that $(Y,\tau {|}_Y)$ is h-$D_1$. □

Remark 8 

(Subspaces). The compatibility hypothesis in Proposition 7 is essential to the argument. In general one must distinguish the topology ${\tau }_h\left(Y\right)$ obtained by applying the h-open construction to $(Y,\tau {|}_Y)$ from the subspace topology ${\tau }_h{\left(X\right)|}_Y$. The proposition gives a useful hereditary theorem precisely when these two constructions agree.

4. Mappings and Preservation Properties

Theorem 4. 

Let $f:X\to Y$ be an h-irresolute surjection. If G is an $hD$-set in Y, then $f^{-1}\left(G\right)$ is an $hD$-set in X.

Proof. 

Write $G=U\setminus V$, where $U,V\in {\tau }_h\left(Y\right)$ and $U\ne Y$. Since f is h-irresolute,

$$f^{-1}\left(U\right),f^{-1}\left(V\right)\in {\tau }_h\left(X\right).$$

Surjectivity and $U\ne Y$ imply $f^{-1}\left(U\right)\ne X$. Therefore

$$f^{-1}\left(G\right)=f^{-1}\left(U\right)\setminus f^{-1}\left(V\right)$$

is an $hD$-set. □

Lemma 2. 

If $f:X\to Y$ is h-continuous and open, then $f^{-1}\left(V\right)$ is h-open in X for every h-open set V in Y.

Proof. 

This is (Theorem 3.9, [1]). □

Corollary 3. 

Let $f:X\to Y$ be an h-continuous open surjection. If G is an $hD$-set in Y, then $f^{-1}\left(G\right)$ is an $hD$-set in X.

Proof. 

By Lemma 2, f is h-irresolute. Apply Theorem 4. □

Theorem 5. 

Let $f:X\to Y$ be an h-irresolute bijection. If Y is h-$D_1$, then X is h-$D_1$.

Proof. 

Let $x\ne y$ in X. Since f is injective, $f\left(x\right)\ne f\left(y\right)$. Choose $hD$-sets $G,E\subseteq Y$ with

$$f\left(x\right)\in G,f\left(y\right)\notin G,f\left(y\right)\in E,f\left(x\right)\notin E.$$

By Theorem 4, $f^{-1}\left(G\right)$ and $f^{-1}\left(E\right)$ are $hD$-sets with the required separation properties. □

Theorem 6. 

A space X is h-$D_1$ if and only if, for every pair of distinct points $x,y\in X$, there exist an h-$D_1$ space Y and an h-irresolute surjection $f:X\to Y$ such that $f\left(x\right)\ne f\left(y\right)$.

Proof. 

If X is h-$D_1$, take $Y=X$ and $f={id}_X$.

Conversely, let $x\ne y$ and choose Y and f as in the hypothesis. Since Y is h-$D_1$, Theorem 1 yields disjoint $hD$-sets $G,E\subseteq Y$ containing $f\left(x\right)$ and $f\left(y\right)$, respectively. Their inverse images are disjoint $hD$-sets in X by Theorem 4; in particular they separate x and y. Hence X is h-$D_1$. □

Theorem 7. 

Let $\{X_i:i\in I\}$ be a family of topological spaces. If every $X_i$ is h-$D_1$, then the product ${\prod }_{i\in I}{X}_i$ is h-$D_1$.

Proof. 

Let $x=\left(x_i\right)$ and $y=\left(y_i\right)$ be distinct points of the product. Choose $j\in I$ with $x_j\ne y_j$. The projection

$$p_j:\prod _{i\in I}{X}_i⟶X_j$$

is a continuous open surjection. By Lemma 2, it is h-irresolute. Since $X_j$ is h-$D_1$ and $p_j\left(x\right)\ne p_j\left(y\right)$, Theorem 6 implies that the product is h-$D_1$. □

5. Concluding Remarks

This paper establishes the associated topology ${\tau }_h$ as the definitive framework for interpreting separation properties built from h-open sets. Within this framework, $hD$-sets are precisely the classical difference sets of $(X,{\tau }_h)$, and the axioms h-$D_1$ and h-$D_2$ become the corresponding $D_1$ and $D_2$ axioms. This viewpoint yields the equivalence h-$D_1\iff h$-$D_2$, gives transparent proofs of the preservation results, and separates genuine interactions with the original topology $\tau$ from statements that are ordinary topological facts in ${\tau }_h$.

The analysis also resolves ambiguities in the surrounding literature. In particular, h-symmetry is identified with the h-$R_0$ condition, while h-$T_1$ is obtained exactly by adjoining h-$T_0$. The h-$R_H$ and h-$R_D$ conditions are stated in terms of singleton h-closures, and Example 3 shows that an h-$R_0$, h-$R_H$ space need not be h-$R_D$ or h-$T_1$. Likewise, Proposition 6 recovers the compact-subset theorem at the correct h-Hausdorff level, whereas Proposition 7 provides a rigorous hereditary result under the natural compatibility condition for associated subspace topologies.

Accordingly, the paper supplies a coherent reference point for future work on h-separation, h-regularity, h-compactness and algebraic structures defined through h-open sets. Further developments should formulate their hypotheses directly in terms of ${\tau }_h$ and explicitly distinguish properties intrinsic to $(X,{\tau }_h)$ from those depending on the comparison between $\tau$ and ${\tau }_h$.