Remarks on h-R_T Topological Spaces

1. Introduction and Preliminaries

The theory of h-open sets was initiated by Abbas [1]. Recall that a topological space is called $T_{1/2}$ if every singleton is either open or closed [4]. Sharma, Saproo, Billawria and Digra subsequently proved that the family of all h-open subsets of an arbitrary topological space forms a topology without requiring the $T_{1/2}$ assumption [10]. This result makes the associated topology the natural setting for the subject.

The closure–kernel, specialization-preorder, and h-$R_0$ framework used below is developed in the companion paper [6], whereas h-$R_0$ was first defined in [7]. Applications to h-regularity are considered in [5], while h-difference separation axioms and the global h-$R_D$ condition are studied in [7]. Related investigations of h-topological groups and h-irresolute topological vector spaces appeared in [8,9]. The present paper forms part of this coordinated series of companion studies and focuses primarily on the pointwise axiom h-$R_T$, which controls the discrepancy between the h-closure and the h-kernel of a singleton.

Two further themes are developed in the associated topology. First, we characterize locally h-indiscrete spaces in terms of h-$R_0$ and the closure of the family of h-open sets under arbitrary intersections. Second, following Di Maio’s weakly $R_0$ axiom [3], we introduce weakly h-$R_0$ spaces, characterize them by singleton h-kernels, and study their behaviour under always h-closed injections and compatible products.

Throughout the paper, $(X,\tau )$ and $(Y,\sigma )$ denote topological spaces, with no separation assumptions unless explicitly stated.

Definition 1

(Classical closure and kernel). Let $(Z,\mu )$ be a topological space and let $A\subseteq Z$. The closure and the kernel of A with respect to μ are

$${Cl}_{\mu }\left(A\right):=\bigcap \{F\subseteq Z:F\mathit{is}\mu \text{-}\mathit{closed}\mathit{and}A\subseteq F\}$$

and

$${Ker}_{\mu }\left(A\right):=\bigcap \{G\in \mu :A\subseteq G\},$$

respectively. Equivalently, $z\in {Cl}_{\mu }\left(A\right)$ if every μ-open neighbourhood of z meets A. When the topology is clear, we write $Cl\left(A\right)$ and $Ker\left(A\right)$.

Definition 2

(Classical separation axioms). Let $(Z,\mu )$ be a topological space.

(1)

The space is $T_0$ if, for every pair of distinct points, an open set contains one of them and not the other.

(2)

The space is $T_1$ if every singleton is closed; equivalently, for every pair of distinct points, each point has an open neighbourhood missing the other.

(3)

The space is $T_{1/2}$ if every singleton is either open or closed [4].

(4)

Following Davis [2], the space is $R_0$ if

$$z\in G\in \mu ⟹{Cl}_{\mu }\left(\left\{z\right\}\right)\subseteq G.$$

Equivalently,

$$z\in {Cl}_{\mu }\left(\left\{w\right\}\right)⟺w\in {Cl}_{\mu }\left(\left\{z\right\}\right)(z,w\in Z).$$

Definition 3

(h-open sets and associated topology). Let $(Z,\mu )$ be a topological space. A subset $A\subseteq Z$ is called h-open with respect to μ if

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

for every nonempty proper set $U\in \mu$. We denote the family of all such sets by ${hO}_{\mu }\left(Z\right)$ and the family of their complements by ${hC}_{\mu }\left(Z\right)$. Sets belonging to both families are called h-clopen. The associated family is denoted by

$${\mu }_h\left(Z\right):={hO}_{\mu }\left(Z\right).$$

Theorem 1

(Sharma et al. [10]). For every topological space $(Z,\mu )$, the family ${\mu }_h\left(Z\right)$ is a topology on Z, and $\mu \subseteq {\mu }_h\left(Z\right)$.

For the spaces $(X,\tau )$ and $(Y,\sigma )$, we write

$${\tau }_h\left(X\right):={hO}_{\tau }\left(X\right),{\sigma }_h\left(Y\right):={hO}_{\sigma }\left(Y\right).$$

When the underlying topology is clear, we abbreviate these as ${\tau }_h$ and ${\sigma }_h$, and we write $hO\left(X\right):={hO}_{\tau }\left(X\right)$ and $hC\left(X\right):={hC}_{\tau }\left(X\right)$. Thus the original topology $\tau$ enters the definition of h-openness through ${Int}_{\tau }$, whereas all closure, kernel and separation constructions carrying the prefix h are the corresponding ordinary constructions in the associated space $(X,{\tau }_h\left(X\right))$.

For $x\in X$, set

$$hO(X,x):=\{U\in hO(X):x\in U\}.$$

The h-closure of $A\subseteq X$ is

$$hCl\left(A\right):=\{x\in X:U\cap A\ne ⌀\mathrm{for}\mathrm{every}U\in hO(X,x\left)\right\}.$$

Equivalently,

$$hCl\left(A\right)=\bigcap \{F\in hC\left(X\right):A\subseteq F\}={Cl}_{{\tau }_h\left(X\right)}\left(A\right).$$

Lemma 1.

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

(1)

If every $A_i$ is h-closed, then ${\bigcap }_{i\in I}{A}_i$ is h-closed.

(2)

If every $A_i$ is h-open, then ${\bigcup }_{i\in I}{A}_i$ is h-open.

(3)

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

(4)

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

(5)

If $A\subseteq B$, then $hCl\left(A\right)\subseteq hCl\left(B\right)$.

(6)

$hCl(hCl(A\left)\right)=hCl\left(A\right)$.

Proof. 

By Theorem 1, these are the ordinary closure properties of the topological space $(X,{\tau }_h\left(X\right))$. □

Definition 4.

For $i\in \{0,1\}$, the space $(X,\tau )$ is called an h-$T_i$ space [7] if the associated space $(X,{\tau }_h\left(X\right))$ satisfies the classical axiom $T_i$ of Definition 2.

2. The h-Kernel and the Specialization Preorder

The terminology and basic facts in this section follow the closure–kernel framework of [6].

Definition 5.

For $A\subseteq X$, the h-kernel of A is

$$hKer\left(A\right):=\bigcap \{G\in hO\left(X\right):A\subseteq G\}={Ker}_{{\tau }_h\left(X\right)}\left(A\right).$$

Lemma 2.

Let $A\subseteq X$ and $x,y\in X$. Then:

(1)

$y\in hKer\left(\right\{x\left\}\right)$ if and only if $x\in hCl\left(\right\{y\left\}\right)$;

(2)

$hKer\left(A\right)=\{x\in X:hCl(\left\{x\right\})\cap A\ne ⌀\}$;

(3)

$hKer\left(\right\{x\left\}\right)\ne hKer\left(\right\{y\left\}\right)$ if and only if $hCl\left(\right\{x\left\}\right)\ne hCl\left(\right\{y\left\}\right)$.

Proof. 

For (1), the condition $y\notin hKer\left(\right\{x\left\}\right)$ means that some h-open set contains x and omits y. This is equivalent to $x\notin hCl\left(\right\{y\left\}\right)$.

For (2), suppose first that $x\in hKer\left(A\right)$ and $hCl\left(\right\{x\left\}\right)\cap A=⌀$. Then $X\setminus hCl\left(\right\{x\left\}\right)$ is an h-open set containing A but not x, contradicting the definition of $hKer\left(A\right)$. Conversely, choose $a\in hCl\left(\right\{x\left\}\right)\cap A$. Every h-open set containing A contains a and, because $a\in hCl\left(\right\{x\left\}\right)$, must also contain x. Hence $x\in hKer\left(A\right)$.

For (3), assume $hCl\left(\right\{x\left\}\right)=hCl\left(\right\{y\left\}\right)$. Then $x\in hCl\left(\right\{y\left\}\right)$ and $y\in hCl\left(\right\{x\left\}\right)$. If $z\in hKer\left(\right\{x\left\}\right)$, item (1) gives $x\in hCl\left(\right\{z\left\}\right)$. By Lemma 1(5)–(6),

$$hCl\left(\right\{y\left\}\right)\subseteq hCl\left(\right\{x\left\}\right)\subseteq hCl\left(\right\{z\left\}\right),$$

so $y\in hCl\left(\right\{z\left\}\right)$ and item (1) gives $z\in hKer\left(\right\{y\left\}\right)$. The reverse inclusion is symmetric. Conversely, equality of the kernels gives $x\in hKer\left(\right\{y\left\}\right)$ and $y\in hKer\left(\right\{x\left\}\right)$; item (1) yields mutual singleton closure membership. Lemma 1(5)–(6) then gives equality of the singleton closures. □

Definition 6.

Recall that a preorder is a reflexive and transitive relation. Theh-specialization preorderon X is defined by

$$x{⪯}_{h}y⟺x\in hCl\left(\left\{y\right\}\right).$$

We write $x{\sim }_{h}y$ when both $x{⪯}_{h}y$ and $y{⪯}_{h}x$, and denote the corresponding equivalence class by ${\left[x\right]}_h$. This is the ordinary specialization preorder of $(X,{\tau }_h\left(X\right))$; compare [11].

Proposition 1.

For all $x,y\in X$, the following statements hold.

(1)

The relation ${⪯}_h$ is a preorder.

(2)

$$hCl\left(\left\{x\right\}\right)=\{y\in X:y{⪯}_{h}x\},hKer\left(\left\{x\right\}\right)=\{y\in X:x{⪯}_{h}y\}.$$

(3)

If $y\in hCl\left(\right\{x\left\}\right)$, then $hCl\left(\right\{y\left\}\right)\subseteq hCl\left(\right\{x\left\}\right)$. Consequently,

$$y\in hCl\left(\right\{x\left\}\right)\mathit{and}x\in hCl\left(\right\{y\left\}\right)⟺hCl\left(\right\{x\left\}\right)=hCl\left(\right\{y\left\}\right).$$

(4)

$${\left[x\right]}_h=hCl\left(\left\{x\right\}\right)\cap hKer\left(\left\{x\right\}\right).$$

(5)

$(X,\tau )$ is h-$T_0$ if and only if ${\left[x\right]}_h=\left\{x\right\}$ for every $x\in X$.

(6)

$(X,\tau )$ is h-$T_1$ if and only if

$$hCl\left(\right\{x\left\}\right)=hKer\left(\right\{x\left\}\right)=\left\{x\right\}(x\in X).$$

Proof. 

Reflexivity follows from $x\in hCl\left(\right\{x\left\}\right)$. If $x{⪯}_{h}y$ and $y{⪯}_{h}z$, then

$$x\in hCl\left(\right\{y\left\}\right)\subseteq hCl(hCl(\left\{z\right\}\left)\right)=hCl\left(\right\{z\left\}\right),$$

by Lemma 1(5)–(6), so $x{⪯}_{h}z$. This proves (1), and (2) follows from Definition 6 and Lemma 2(1).

If $y{⪯}_{h}x$, then $\left\{y\right\}\subseteq hCl\left(\right\{x\left\}\right)$, and Lemma 1(5)–(6) gives $hCl\left(\right\{y\left\}\right)\subseteq hCl\left(\right\{x\left\}\right)$, proving (3). Mutual membership gives both inclusions. Statement (4) is immediate from (2). Finally, (5) and (6) are exactly the classical $T_0$ and $T_1$ characterizations of the associated space, using Definitions 2 and 4. □

Definition 7.

A space $(X,\tau )$ is called h-$R_0$ [7] if every h-open set contains the h-closure of each of its points. Equivalently,

$$x\in hCl\left(\right\{y\left\}\right)⟺y\in hCl\left(\right\{x\left\}\right)(x,y\in X).$$

Equivalently again, $(X,{\tau }_h\left(X\right))$ is an ordinary $R_0$ space in the sense of Definition 2; see also [6].

Proposition 2.

For a topological space $(X,\tau )$, the following conditions are equivalent:

(1)

$(X,\tau )$ is h-$R_0$;

(2)

$hCl\left(\right\{x\left\}\right)\subseteq hKer\left(\right\{x\left\}\right)$ for every $x\in X$;

(3)

$hKer\left(\right\{x\left\}\right)\subseteq hCl\left(\right\{x\left\}\right)$ for every $x\in X$;

(4)

$hCl\left(\right\{x\left\}\right)=hKer\left(\right\{x\left\}\right)$ for every $x\in X$.

Proof. 

By Definition 7, condition (1) is equivalent to symmetry of singleton closure membership. Using Lemma 2(1), condition (2) says that

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

for all $x,y\in X$, and is therefore equivalent to (1). Similarly, condition (3) says that

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

and is also equivalent to (1). Conditions (2) and (3) together are equivalent to the equality in (4). □

Remark 1.

Proposition 2 gives an entirely set-theoretic characterization: $(X,\tau )$ is h-$R_0$ precisely when, for every $x\in X$, the intersection of all h-open sets containing x coincides with the intersection of all h-closed sets containing x.

3. h-$R_T$ Spaces

We shall use the following terminology. The global h-$R_D$ condition is the one adopted in the companion study [7]; the conditional notation h-$R_D^{*}$ is introduced here to distinguish it from that global condition. To avoid terminological ambiguity across the companion papers, we reserve the symbol h-$R_H$ exclusively for the closure-intersection property in Definition 8(1), while the term h-semisimple is used exclusively for the openness of singleton h-closures in Definition 8(2).

Definition 8.

A topological space $(X,\tau )$ is called

(1)

h-$R_H$ [7] if, for all $x,y\in X$,

$$hCl\left(\right\{x\left\}\right)\ne hCl\left(\right\{y\left\}\right)⟹hCl\left(\right\{x\left\}\right)\cap hCl\left(\right\{y\left\}\right)\in \{⌀,\{x\},\{y\left\}\right\};$$

(2)

h-semisimple if $hCl\left(\right\{x\left\}\right)$ is h-open for every $x\in X$;

(3)

h-$R_D$ [7] if the h-derived set

$$hDer\left(\right\{x\left\}\right):=hCl\left(\right\{x\left\}\right)\setminus \left\{x\right\}$$

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

(4)

h-$R_D^{*}$ if, for every $x\in X$ satisfying

$$hCl\left(\right\{x\left\}\right)\cap hKer\left(\right\{x\left\}\right)=\left\{x\right\},$$

the set $hDer\left(\right\{x\left\}\right)$ is h-closed.

The global condition h-$R_D$ implies h-$R_D^{*}$, whereas the converse need not hold.

Definition 9.

A set is called degenerate if it contains at most one point.

Definition 10.

A topological space $(X,\tau )$ is called an h-$R_T$ space if, for every $x\in X$, both sets

$$hKer\left(\right\{x\left\}\right)\setminus hCl\left(\right\{x\left\}\right)\mathrm{and}hCl\left(\right\{x\left\}\right)\setminus hKer\left(\right\{x\left\}\right)$$

are degenerate in the sense of Definition 9.

Proposition 3

(Order-theoretic characterization). For $x\in X$, set

$$\begin{array}{cc}\hfill L_h\left(x\right)& :=\{y\in X:y{⪯}_{h}x\mathrm{and}{x}_{h}y\},\hfill \\ \hfill U_h\left(x\right)& :=\{y\in X:x{⪯}_{h}y\mathrm{and}{y}_{h}x\}.\hfill \end{array}$$

Then

$$L_h\left(x\right)=hCl\left(\left\{x\right\}\right)\setminus hKer\left(\left\{x\right\}\right),U_h\left(x\right)=hKer\left(\left\{x\right\}\right)\setminus hCl\left(\left\{x\right\}\right).$$

Consequently, $(X,\tau )$ is h-$R_T$ if and only if

$$|L_h\left(x\right)|\le 1\mathrm{and}|U_h\left(x\right)|\le 1(x\in X).$$

Proof. 

The identities follow directly from Proposition 1(2). The final assertion is exactly Definition 10 expressed in the h-specialization preorder. □

Every h-$R_0$ space is h-$R_T$, since both differences in Definition 10 are then empty. The converse fails.

Example 1

(h-$R_T$ does not imply h-$R_0$). Let

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

A direct computation gives

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

The singleton h-closures and h-kernels are

Thus the only nonempty differences in Definition 10 are

$$hCl\left(\right\{b\left\}\right)\setminus hKer\left(\right\{b\left\}\right)=\left\{c\right\},hKer\left(\right\{c\left\}\right)\setminus hCl\left(\right\{c\left\}\right)=\left\{b\right\}.$$

Hence the space is h-$R_T$. It is not h-$R_0$, because $hCl\left(\right\{b\left\}\right)\ne hKer\left(\right\{b\left\}\right)$.

Example 2

(h-$T_0$ does not imply h-$R_T$). Let

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

Here ${\tau }_h\left(X\right)=\tau$. The associated space is $T_0$, hence $(X,\tau )$ is h-$T_0$. On the other hand,

$$hCl\left(\right\{c\left\}\right)=\left\{c\right\},hKer\left(\right\{c\left\}\right)=X,$$

and therefore

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

which is not degenerate. Thus the space is not h-$R_T$.

Moreover, this space is h-$R_D$, since

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

so every singleton h-derived set is either $\left\{c\right\}$ or ⌀, and both are h-closed. Hence h-$R_D$ does not imply h-$R_T$.

Example 3.

Let

$$X=\{a,b,c\},\tau =\{⌀,\{b\},X\}.$$

Then

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

Consequently,

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

and the corresponding h-kernels coincide with these closures. Hence the space is h-$R_0$, and therefore h-$R_T$. It is also h-$R_H$. Moreover,

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

so every singleton h-closure is h-open and the space is h-semisimple.

However, it is not h-$R_D$. Indeed,

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

and $\left\{c\right\}$ is not h-closed because its complement $\{a,b\}$ is not h-open. The space nevertheless satisfies h-$R_D^{*}$: for a and c the antecedent in Definition 8(4) fails, while for b the derived set is empty. The original topology $(X,\tau )$ is not $R_0$, since ${Cl}_{\tau }\left(\left\{b\right\}\right)=X\ne \left\{b\right\}$.

Remark 2.

Example 3 shows that three singleton conditions considered in this paper describe genuinely different features. Every singleton h-closure is h-open, so the space is h-semisimple. It is also h-$R_0$, and hence h-$R_T$, because the singleton h-closures and h-kernels coincide. Nevertheless,

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

is not h-closed. Thus openness of singleton closures, control of the closure–kernel discrepancy, and closedness of punctured singleton closures concern different aspects of the local specialization structure. In particular, neither h-semisimplicity alone nor even its conjunction with h-$R_0$ guarantees the global h-$R_D$ condition.

Theorem 2.

Every h-$R_T$ space is h-$R_D^{*}$.

Proof. 

Let $x\in X$ and assume

$$hCl\left(\right\{x\left\}\right)\cap hKer\left(\right\{x\left\}\right)=\left\{x\right\}.$$

Set

$$D:=hCl\left(\right\{x\left\}\right)\setminus hKer\left(\right\{x\left\}\right).$$

By Definition 10, the set D is degenerate in the sense of Definition 9; hence it is either empty or a singleton. The displayed intersection gives

$$hDer\left(\right\{x\left\}\right)=hCl\left(\right\{x\left\}\right)\setminus \left\{x\right\}=D.$$

If $D=⌀$, it is h-closed. Suppose $D=\left\{d\right\}$. Since $d\notin hKer\left(\right\{x\left\}\right)$, there exists an h-open set V such that $x\in V$ and $d\notin V$. As $hCl\left(\right\{x\left\}\right)=\{x,d\}$,

$$D=hCl\left(\right\{x\left\}\right)\cap (X\setminus V).$$

Both sets on the right are h-closed; hence D is h-closed. □

Corollary 1.

Every h-$T_0$, h-$R_T$ space is h-$R_D$.

Proof. 

By Proposition 1(4)–(5), the h-$T_0$ condition gives

$$hCl\left(\left\{x\right\}\right)\cap hKer\left(\left\{x\right\}\right)={\left[x\right]}_h=\left\{x\right\}(x\in X).$$

Theorem 2 therefore applies to every point, which is exactly the global h-$R_D$ condition. □

Theorem 3.

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

Proof. 

Let $x,y\in X$ satisfy

$$hCl\left(\right\{x\left\}\right)\ne hCl\left(\right\{y\left\}\right).$$

Then $x\ne y$. Suppose, toward a contradiction, that there exists $a\in hCl\left(\right\{x\left\}\right)\cap hCl\left(\right\{y\left\}\right)$ with $a\notin \{x,y\}$. Lemma 2(1) gives

$$x,y\in hKer\left(\right\{a\left\}\right).$$

Set

$$P_a:=hCl\left(\left\{a\right\}\right)\cap hKer\left(\left\{a\right\}\right),E_a:=hKer\left(\left\{a\right\}\right)\setminus hCl\left(\left\{a\right\}\right).$$

By Definition 10, $E_a=hKer\left(\left\{a\right\}\right)\setminus hCl\left(\left\{a\right\}\right)$ is degenerate in the sense of Definition 9. Since $x\ne y$, at least one of $x,y$ lies in $P_a$. Without loss of generality, let $y\in P_a$. Then

$$y\in hCl\left(\right\{a\left\}\right)\mathrm{and}a\in hCl\left(\right\{y\left\}\right),$$

so Proposition 1(3) yields $hCl\left(\right\{a\left\}\right)=hCl\left(\right\{y\left\}\right)$.

If $x\in P_a$, the same proposition gives $hCl\left(\right\{x\left\}\right)=hCl\left(\right\{a\left\}\right)$, a contradiction. Thus $x\in E_a$, so $x\notin hCl\left(\right\{a\left\}\right)$. Since $hCl\left(\right\{a\left\}\right)$ is h-closed, its complement is an h-open neighbourhood of x, and hence

$$hKer\left(\right\{x\left\}\right)\subseteq X\setminus hCl\left(\right\{a\left\}\right).$$

On the other hand, $a\in hCl\left(\right\{x\left\}\right)$ implies, by Proposition 1(3), that

$$hCl\left(\right\{a\left\}\right)\subseteq hCl\left(\right\{x\left\}\right).$$

Since $y\in hCl\left(\right\{a\left\}\right)$, we obtain

$$\{a,y\}\subseteq hCl\left(\right\{x\left\}\right)\setminus hKer\left(\right\{x\left\}\right),$$

contradicting the h-$R_T$ condition. Therefore

$$hCl\left(\right\{x\left\}\right)\cap hCl\left(\right\{y\left\}\right)\in \{⌀,\{x\},\{y\left\}\right\}.$$

□

Remark 3.

The axiom h-$R_T$ lies between exact specialization symmetry and weaker singleton conditions. The h-$R_0$ property forces $hCl\left(\right\{x\left\}\right)=hKer\left(\right\{x\left\}\right)$, whereas h-$R_T$ permits at most one strict predecessor and at most one strict successor of each point outside its indistinguishability class. This controlled defect is sufficient for h-$R_D^{*}$ and h-$R_H$, and, under h-$T_0$, for the global h-$R_D$ property.

Corollary 2.

The following implications hold:

$$h\text{-}{R}_0⟹h\text{-}{R}_T⟹h\text{-}{R}_D^{*},h\text{-}{R}_T⟹h\text{-}{R}_H,h\text{-}{R}_D⟹h\text{-}{R}_D^{*},$$

and

$$h\text{-}{T}_0+h\text{-}{R}_T⟹h\text{-}{R}_D.$$

The implication h-$R_0\Rightarrow h$-$R_T$ is not reversible; h-$T_0$ does not imply h-$R_T$; and the global conditions h-$R_D$ and h-$R_T$ are incomparable.

4. Subspaces and Finite Products

For a subset $Y\subseteq X$, let

$${\tau |}_Y:=\{U\cap Y:U\in \tau \}$$

be the original subspace topology. We write

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

for the associated topology constructed from $(Y,\tau {|}_Y)$, whereas

$${\tau }_h{\left(X\right)|}_Y:=\{H\cap Y:H\in {\tau }_h\left(X\right)\}$$

denotes the trace on Y of the associated topology of X. The operators ${hCl}_X$ and ${hKer}_X$ are computed in $(X,{\tau }_h\left(X\right))$, while ${hCl}_Y$ and ${hKer}_Y$ are computed in $(Y,{\tau }_h\left(Y\right))$.

The equality ${\tau }_h\left(Y\right)={\tau }_h{\left(X\right)|}_Y$ is not automatic. Under this compatibility condition, however, the singleton closure–kernel structure restricts correctly.

Proposition 4.

Let $(X,\tau )$ be an h-$R_T$ 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-$R_T$.

Figure 1. Logical relations among the principal h-separation properties considered here. Solid arrows denote valid implications. The arrow from h-$R_0$ to weakly h-$R_0$ requires $\left|X\right|\ne 1$; its converse fails by Example 1. Dashed slashed arrows denote implications that fail in general, and the dashed two-headed slashed segment records incomparability. The double-bordered node emphasizes the joint hypothesis in Corollary 1. Local h-indiscreteness is characterized separately in Theorem 4.

Figure 1. Logical relations among the principal h-separation properties considered here. Solid arrows denote valid implications. The arrow from h-$R_0$ to weakly h-$R_0$ requires $\left|X\right|\ne 1$; its converse fails by Example 1. Dashed slashed arrows denote implications that fail in general, and the dashed two-headed slashed segment records incomparability. The double-bordered node emphasizes the joint hypothesis in Corollary 1. Local h-indiscreteness is characterized separately in Theorem 4.

Proof. 

Under the compatibility assumption, ordinary subspace closure and kernel in the associated topology give, for every $y\in Y$,

$${hCl}_Y\left(\left\{y\right\}\right)={hCl}_X\left(\left\{y\right\}\right)\cap Y,{hKer}_Y\left(\left\{y\right\}\right)={hKer}_X\left(\left\{y\right\}\right)\cap Y.$$

Hence

$${hKer}_Y\left(\left\{y\right\}\right)\setminus {hCl}_Y\left(\left\{y\right\}\right)=\left({hKer}_X\left(\left\{y\right\}\right)\setminus {hCl}_X\left(\left\{y\right\}\right)\right)\cap Y,$$

and similarly

$${hCl}_Y\left(\left\{y\right\}\right)\setminus {hKer}_Y\left(\left\{y\right\}\right)=\left({hCl}_X\left(\left\{y\right\}\right)\setminus {hKer}_X\left(\left\{y\right\}\right)\right)\cap Y.$$

Each set is a subset of a degenerate set and is therefore degenerate. □

Remark 4.

Proposition 4 shows that h-$R_T$ is hereditary under compatible subspaces. The compatibility hypothesis is essential because the associated topology constructed from the original subspace topology need not equal the trace of the associated topology of the ambient space.

Example 4

(The compatibility condition is not automatic). Let

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

Then

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

The original subspace topology ${\tau |}_Y$ is indiscrete. Since it has no nonempty proper open set, every subset of Y is h-open, and hence

$${\tau }_h\left(Y\right)=\mathcal{P}\left(Y\right)\ne {\tau }_h{\left(X\right)|}_Y.$$

Proposition 5.

Let $\{(X_i,{\tau }_i):1\le i\le n\}$ be a finite family of nonempty topological spaces. Let

$$X:={\displaystyle \prod _{i=1}^n}{X}_i$$

be endowed with the product topology τ generated by the original topologies ${\tau }_i$, and set

$${\tau }_{i,h}:={hO}_{{\tau }_i}\left(X_i\right)(1\le i\le n).$$

Assume that the associated topology ${\tau }_h\left(X\right)={hO}_{\tau }\left(X\right)$ coincides with the product topology on X generated by the factor topologies ${\tau }_{i,h}$. Then:

(1)

if every $(X_i,{\tau }_i)$ is h-$R_0$, then $(X,\tau )$ is h-$R_0$, and hence h-$R_T$;

(2)

if one factor $(X_j,{\tau }_j)$ is h-$R_T$ and every factor $(X_i,{\tau }_i)$ with $i\ne j$ is h-$T_1$, then $(X,\tau )$ is h-$R_T$.

Proof. 

Let ${hCl}_i$ and ${hKer}_i$ denote closure and kernel in the associated factor space $(X_i,{\tau }_{i,h})$. Under the stated compatibility condition, for $x=(x_1,\dots ,x_n)\in X$ one has

$${hCl}_X\left(\left\{x\right\}\right)={\displaystyle \prod _{i=1}^n}{hCl}_i\left(\left\{x_i\right\}\right),{hKer}_X\left(\left\{x\right\}\right)={\displaystyle \prod _{i=1}^n}{hKer}_i\left(\left\{x_i\right\}\right).$$

If every factor is h-$R_0$, the corresponding closure and kernel agree in each coordinate, proving (1).

For (2), Proposition 1(6), applied to each associated factor space, gives

$${hCl}_i\left(\left\{x_i\right\}\right)={hKer}_i\left(\left\{x_i\right\}\right)=\left\{x_i\right\}(i\ne j).$$

Therefore the two set differences in the product are naturally in bijection with

$${hCl}_j\left(\left\{x_j\right\}\right)\setminus {hKer}_j\left(\left\{x_j\right\}\right)\mathrm{and}{hKer}_j\left(\left\{x_j\right\}\right)\setminus {hCl}_j\left(\left\{x_j\right\}\right),$$

respectively. Each contains at most one point because the j-th factor is h-$R_T$. □

Remark 5.

The product compatibility condition in Proposition 5 must be verified; it does not follow merely from the hypotheses on the factor spaces. No unconditional preservation theorem for arbitrary finite or infinite products is claimed.

5. Locally h-Indiscrete Spaces

Definition 11.

A topological space $(X,\tau )$ is calledlocally h-indiscreteif, for every $x\in X$, there exists a set $N_x\in {\tau }_h\left(X\right)$ such that $x\in N_x$ and the subspace

$$(N_x,{\tau }_h\left(X\right){|}_{N_x})$$

is indiscrete.

Theorem 4.

A topological space $(X,\tau )$ is locally h-indiscrete if and only if the following conditions hold:

(1)

$(X,\tau )$ is h-$R_0$;

(2)

the intersection of every family of h-open subsets of X is h-open.

Proof. 

Assume first that $(X,\tau )$ is locally h-indiscrete. Let $x\in U\in {\tau }_h\left(X\right)$ and let $y\in hCl\left(\right\{x\left\}\right)$. Choose an h-open neighbourhood $N_y$ of y whose induced topology is indiscrete. Since $y\in hCl\left(\right\{x\left\}\right)$, every h-open neighbourhood of y contains x, and hence $x\in N_y$. The set $U\cap N_y$ is a nonempty open subset of the indiscrete subspace $N_y$, so $U\cap N_y=N_y$. Thus $y\in U$, and consequently $hCl\left(\right\{x\left\}\right)\subseteq U$. This proves that $(X,\tau )$ is h-$R_0$.

Let $\{U_i:i\in I\}$ be a family of h-open subsets of X, and put $A:={\bigcap }_{i\in I}{U}_i$. If $A=⌀$, then A is h-open. Otherwise, for each $x\in A$, choose an h-open indiscrete neighbourhood $N_x$. For every $i\in I$, the set $U_i\cap N_x$ is a nonempty open subset of $N_x$, and hence equals $N_x$. Therefore $N_x\subseteq A$, and

$$A=\bigcup _{x\in A}{N}_x$$

is h-open. This proves condition (2).

Conversely, assume conditions (1) and (2). For $x\in X$, define

$$N_x:=hKer\left(\left\{x\right\}\right)=\bigcap \{U\in {\tau }_h\left(X\right):x\in U\}.$$

By condition (2), $N_x$ is h-open, and Proposition 2 gives

$$N_x=hCl\left(\left\{x\right\}\right).$$

Let $U\in {\tau }_h\left(X\right)$ and suppose that $U\cap N_x\ne ⌀$. Choose $y\in U\cap N_x$. Since $y\in hCl\left(\right\{x\left\}\right)$ and the space is h-$R_0$, one has $x\in hCl\left(\right\{y\left\}\right)$ and hence, by Proposition 1(3),

$$hCl\left(\left\{y\right\}\right)=hCl\left(\left\{x\right\}\right)=N_x.$$

The h-$R_0$ property and $y\in U$ imply $hCl\left(\right\{y\left\}\right)\subseteq U$, so $N_x\subseteq U$. Thus every nonempty open subset of the subspace $N_x$ is all of $N_x$, and $(N_x,{\tau }_h\left(X\right){|}_{N_x})$ is indiscrete. Therefore $(X,\tau )$ is locally h-indiscrete. □

6. Weakly h-$R_0$ Spaces

Di Maio introduced the notion of weakly $R_0$ in 1985 [3]. In that paper, the topologies under consideration are assumed to be strictly finer than the trivial topology. A topological space $(X,\tau )$ is weakly $R_0$ if

$$\bigcap _{x\in X}{Cl}_{\tau }\left(\left\{x\right\}\right)=⌀.$$

The associated-topology analogue is the following.

Definition 12.

A topological space $(X,\tau )$ is calledweakly h-$R_0$if

$$\bigcap _{x\in X}hCl\left(\left\{x\right\}\right)=⌀.$$

Remark 6.

Since $\tau \subseteq {\tau }_h\left(X\right)$, one has $hCl\left(A\right)\subseteq {Cl}_{\tau }\left(A\right)$ for every $A\subseteq X$. Hence every weakly $R_0$ space is weakly h-$R_0$.

Except for the one-point space, every h-$R_0$ space is weakly h-$R_0$. Indeed, if $\left|X\right|\ge 2$, then ${\tau }_h\left(X\right)$ is not indiscrete: either τ contains a nonempty proper open set, which also belongs to ${\tau }_h\left(X\right)$, or τ is indiscrete, in which case the defining condition of h-openness is vacuous and ${\tau }_h\left(X\right)=\mathcal{P}\left(X\right)$. In an $R_0$ space the closures of singletons are the equivalence classes of topological indistinguishability. Since ${\tau }_h\left(X\right)$ is not indiscrete, there are at least two such classes, and their total intersection is empty. For a one-point space, by contrast, the unique singleton h-closure is X, so the space is h-$R_0$ but not weakly h-$R_0$.

The converses fail. In Example 1,

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

and hence the intersection of the singleton h-closures is empty. Thus the space is weakly h-$R_0$, although it is not h-$R_0$. With respect to the original topology,

$${Cl}_{\tau }\left(\left\{a\right\}\right)=X,{Cl}_{\tau }\left(\left\{b\right\}\right)=\{b,c\},{Cl}_{\tau }\left(\left\{c\right\}\right)=\left\{c\right\},$$

so their intersection is $\left\{c\right\}$; consequently, the space is not weakly $R_0$.

Theorem 5.

A topological space $(X,\tau )$ is weakly h-$R_0$ if and only if

$$hKer\left(\right\{x\left\}\right)\ne X\mathrm{for}\mathrm{every}x\in X.$$

Proof. 

For every $y\in X$, Lemma 2(1) gives

$$\begin{array}{cc}\hfill y\in \bigcap _{x\in X}hCl\left(\left\{x\right\}\right)& ⟺y\in hCl\left(\right\{x\left\}\right)\mathrm{for}\mathrm{every}x\in X\hfill \\ \hfill & ⟺x\in hKer\left(\right\{y\left\}\right)\mathrm{for}\mathrm{every}x\in X\hfill \\ \hfill & ⟺hKer\left(\right\{y\left\}\right)=X.\hfill \end{array}$$

Therefore

$$\bigcap _{x\in X}hCl\left(\left\{x\right\}\right)=⌀$$

if and only if no point $y\in X$ satisfies $hKer\left(\right\{y\left\}\right)=X$. □

Definition 13.

A function $f:(X,\tau )\to (Y,\sigma )$ is calledalways h-closedif $f\left(F\right)$ is h-closed in $(Y,\sigma )$ whenever F is h-closed in $(X,\tau )$.

Theorem 6.

Let X be nonempty. If $f:(X,\tau )\to (Y,\sigma )$ is an injective always h-closed function and $(X,\tau )$ is weakly h-$R_0$, then $(Y,\sigma )$ is weakly h-$R_0$.

Proof. 

Write ${hCl}_X$ and ${hCl}_Y$ for the h-closure operators in the source and target spaces. For every $x\in X$, the set $f\left({hCl}_X\left(\left\{x\right\}\right)\right)$ is h-closed in Y and contains $f\left(x\right)$. Hence

$${hCl}_Y\left(\left\{f\left(x\right)\right\}\right)\subseteq f\left({hCl}_X\left(\left\{x\right\}\right)\right).$$

Consequently,

$$\begin{array}{cc}\hfill \bigcap _{y\in Y}{hCl}_Y\left(\left\{y\right\}\right)& \subseteq \bigcap _{x\in X}{hCl}_Y\left(\left\{f\left(x\right)\right\}\right)\hfill \\ \hfill & \subseteq \bigcap _{x\in X}f\left({hCl}_X\left(\left\{x\right\}\right)\right)\hfill \\ \hfill & =f\left(\bigcap _{x\in X}{hCl}_X\left(\left\{x\right\}\right)\right)\hfill \\ \hfill & =f(⌀)=⌀.\hfill \end{array}$$

The equality in the third line uses the injectivity of f. □

Theorem 7

(Compatible product preservation). Let $(X,\tau )$ and $(Y,\sigma )$ be nonempty topological spaces, and endow $X\times Y$ with the product topology $\tau \times \sigma$. Assume that

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

where the right-hand side denotes the product of the associated topologies. If $(X,\tau )$ is weakly h-$R_0$, then $(X\times Y,\tau \times \sigma )$ is weakly h-$R_0$.

Proof. 

Under the compatibility hypothesis,

$${hCl}_{X\times Y}\left(\left\{(x,y)\right\}\right)={hCl}_X\left(\left\{x\right\}\right)\times {hCl}_Y\left(\left\{y\right\}\right)((x,y)\in X\times Y).$$

Therefore

$$\begin{array}{cc}\hfill \bigcap _{(x,y)\in X\times Y}{hCl}_{X\times Y}\left(\left\{(x,y)\right\}\right)& =\left(\bigcap _{x\in X}{hCl}_X\left(\left\{x\right\}\right)\right)\times \left(\bigcap _{y\in Y}{hCl}_Y\left(\left\{y\right\}\right)\right)\hfill \\ \hfill & =⌀.\hfill \end{array}$$

□

Remark 7

(The compatibility hypothesis is essential). Theorem 7 is false without the compatibility assumption. Let

$$X=\{0,1\},\tau =\{⌀,X\}.$$

Since τ has no nonempty proper open set, ${\tau }_h\left(X\right)=\mathcal{P}\left(X\right)$, and hence $(X,\tau )$ is weakly h-$R_0$. Let

$$Y=\{a,b,c\},\sigma =\{⌀,\{a\},\{b\},\{a,b\},Y\}.$$

The product topology on $X\times Y$ is

$$\{⌀,X\times \{a\},X\times \{b\},X\times \{a,b\},X\times Y\},$$

and a direct computation shows that it coincides with its associated topology. For $i\in \{0,1\}$,

$$\begin{array}{cc}\hfill hCl\left(\right\{(i,a)\left\}\right)& =X\times \{a,c\},\hfill \\ \hfill hCl\left(\right\{(i,b)\left\}\right)& =X\times \{b,c\},\hfill \\ \hfill hCl\left(\right\{(i,c)\left\}\right)& =X\times \left\{c\right\}.\hfill \end{array}$$

Thus

$$\bigcap _{(x,y)\in X\times Y}hCl\left(\left\{(x,y)\right\}\right)=X\times \left\{c\right\}\ne ⌀,$$

so $X\times Y$ is not weakly h-$R_0$. In this example,

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

7. Concluding Remarks

The associated topology ${\tau }_h\left(X\right)$ is the natural framework for the axioms studied in this paper. The h-$R_0$ property is precisely the classical $R_0$ condition of $(X,{\tau }_h\left(X\right))$, whereas h-$R_T$ controls, point by point, the strict lower and upper specialization sets outside the indistinguishability class ${\left[x\right]}_h$.

The main results establish

$$h\text{-}{R}_0⟹h\text{-}{R}_T⟹h\text{-}{R}_D^{*},h\text{-}{R}_T⟹h\text{-}{R}_H,$$

and show that the additional h-$T_0$ hypothesis yields the global h-$R_D$ condition. The finite examples prove that h-$R_T$ does not imply h-$R_0$, that h-$T_0$ does not imply h-$R_T$, and that the global conditions h-$R_T$ and h-$R_D$ are incomparable.

The local structure of the associated topology is further clarified by Theorem 4: local h-indiscreteness is equivalent to the conjunction of h-$R_0$ and the Alexandrov-type condition that arbitrary intersections of h-open sets remain h-open. This identifies the singleton h-kernel as an h-open indiscrete neighbourhood of each point.

Following Di Maio, we introduced the weakly h-$R_0$ condition. Theorem 5 characterizes it by the absence of a point whose singleton h-kernel is all of X. The property is preserved by injective always h-closed maps from a nonempty domain and by products satisfying the compatibility condition of Theorem 7. Remark 7 shows that this compatibility cannot be omitted. Example 1 also demonstrates that weakly h-$R_0$ is strictly weaker than both h-$R_0$ and the classical weakly $R_0$ condition.

The present paper forms part of a coordinated series of companion studies based on the associated topology. The preservation results identify the compatibility assumptions needed for subspaces and products, while the specialization-preorder and singleton-kernel formulations supply a reusable framework for further work on mappings and algebraic topological structures; see also [8,9].