Topology, Fineness and Separation Axioms for h-Open Sets in Relator Spaces

1. Introduction

A relator on a set X is a family of binary relations on X. The pair $(X,\mathcal{R})$ is called a relator space. This approach was developed by Száz as a common relational framework for topology, uniform structures and related generalized structures [8,9]. In particular, relator spaces allow topological notions to be formulated in terms of families of relations rather than only in terms of neighborhoods. For a relator $\mathcal{R}$ on X, one naturally defines the relator interior by

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

The associated family

$${\tau }_{\mathcal{R}}=\{A\subseteq X:A\subseteq {int}_{\mathcal{R}}\left(A\right)\}$$

is the natural family of relator-open sets. The systematic study of generalized topologically open sets in relator spaces has been developed by Rassias, Salih and Száz [5], who also emphasized the role of Pervin-type preorder relations. The present paper introduces a second layer of openness. Following the theory of h-open sets, we say that a subset A is h-open when its openness is stable under adjoining a nonempty proper open set and then taking relator interior. In the classical topological setting the corresponding family of h-open sets has recently been studied as an associated topology

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

See Abbas [1] and the subsequent clarification of the topological structure of the family of h-open sets [3,7]. Our aim is to transfer these ideas to arbitrary relator spaces. The main questions are the following.

Question 1.

Characterize those relators $\mathcal{R}$ for which $hO(X,\mathcal{R})$ is always a topology.

Question 2.

Characterize the relators for which $hO(X,\mathcal{R})={\tau }_{\mathcal{R}}$.

Question 3.

Determine conditions under which $hO(X,\mathcal{R})$ is strictly finer than ${\tau }_{\mathcal{R}}$.

Question 4.

Develop h-versions of the separation axioms $T_0,T_1,T_2,R_0,R_1$ and regularity directly for relator spaces.

The first three questions concern the set-theoretic structure of the family of h-open sets, while the fourth concerns the separation theory induced by that family.

2. Relator Spaces and Relator-Open Sets

Definition 1.

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

For $R\in \mathcal{R}$ and $x\in X$, put $R\left(x\right)=\{y\in X:(x,y)\in R\}$. For $A\subseteq X$, define $R\left[A\right]={\bigcup }_{x\in A}R\left(x\right)$. The following interior operator is fundamental in the theory of relator spaces [5,8].

Definition 2.

For $A\subseteq X$, define

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

Definition 3.

A subset $A\subseteq X$ is calledrelator-openif

$$A\subseteq {int}_{\mathcal{R}}\left(A\right).$$

The family of all relator-open subsets is denoted by

$${\tau }_{\mathcal{R}}=\{A\subseteq X:A\subseteq {int}_{\mathcal{R}}\left(A\right)\}.$$

The notation ${\tau }_{\mathcal{R}}$ is standard in the literature on relator spaces and generalized topologically open sets [5].

Remark 1.

In general, the family ${\tau }_{\mathcal{R}}$ should initially be regarded as a family of relator-open sets. Additional structural conditions on $\mathcal{R}$ guarantee that it satisfies the topological axioms. Pervin-type preorder relators provide an important source of examples associated with generalized topologies [5].

3. h-Open Subsets of Relator Spaces

We now formulate the h-open concept directly in terms of a relator.

Definition 4.

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

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

for every nonempty proper relator-open set

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

The family of all h-open subsets of $(X,\mathcal{R})$ is denoted by

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

This definition reduces to the usual definition of an h-open subset when $\mathcal{R}$ is a topological relator. In the topological setting, the family of h-open sets has been studied as an associated topology [1,3,7].

Proposition 1.

For every relator space $(X,\mathcal{R})$,

$$⌀,X\in hO(X,\mathcal{R}).$$

Proof. 

The empty set satisfies the defining condition trivially. For X, let $V\in {\tau }_{\mathcal{R}}$ be nonempty and proper. Since $X\cup V=X$, we have

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

whenever $\mathcal{R}$ contains a relation whose sections contain the relevant points. In particular, under the standard reflexive relator assumption, $X\subseteq {int}_{\mathcal{R}}\left(X\right)$, and hence X is h-open. □

Remark 2.

The reflexivity assumption is natural in many applications of relators. When reflexivity is not assumed, the inclusion $X\in hO(X,\mathcal{R})$ should be imposed explicitly or obtained from an appropriate covering condition on $\mathcal{R}$.

4. When Is $hO(X,\mathcal{R})$ a Topology?

The first problem is to determine conditions guaranteeing closure under arbitrary unions and finite intersections.

Definition 5.

We say that a relator $\mathcal{R}$ has theh-union propertyif, for every family $\{A_i:i\in I\}$ of h-open sets,

$$\bigcup _{i\in I}{A}_i$$

is h-open.

Definition 6.

We say that $\mathcal{R}$ has theh-intersection propertyif for all $A,B\in hO(X,\mathcal{R})$,

$$A\cap B\in hO(X,\mathcal{R}).$$

The next proposition gives a direct characterization.

Proposition 2.

Assume that $\mathcal{R}$ is such that $X\in hO(X,\mathcal{R})$. Then $hO(X,\mathcal{R})$ is a topology on X if and only if $\mathcal{R}$ has both the h-union property and the h-intersection property.

Proof. 

The necessity follows immediately from the axioms of a topology. Conversely, if the two properties hold, then $⌀,X\in hO(X,\mathcal{R})$, arbitrary unions of h-open sets are h-open, and finite intersections of h-open sets are h-open. Hence $hO(X,\mathcal{R})$ is a topology. □

The condition can be expressed directly in terms of the relator interior.

Theorem 5.

Suppose that ${\tau }_{\mathcal{R}}$ is a topology. If for all $A,B\subseteq X$ and all nonempty proper $V\in {\tau }_{\mathcal{R}}$,

$${int}_{\mathcal{R}}(A\cup V)\cap {int}_{\mathcal{R}}(B\cup V)\subseteq {int}_{\mathcal{R}}((A\cap B)\cup V),$$

(4.1)

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

Proof. 

Let $A,B\in hO(X,\mathcal{R})$ and let $V\in {\tau }_{\mathcal{R}}$ be nonempty and proper. Since A and B are h-open,

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

and

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

Therefore

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

By (4.1),

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

Hence $A\cap B$ is h-open. Since ${\tau }_{\mathcal{R}}$ is a topology, the usual union argument together with the defining condition for h-openness gives closure under arbitrary unions. Thus $hO(X,\mathcal{R})$ is a topology. □

Remark 3.

Condition (4.1) is a useful sufficient condition rather than a claim that it is necessary for every relator. It isolates the precise interaction between the relator interior and finite intersections that is needed in the proof.

5. A Sufficient Relational Condition

The next result gives a more concrete sufficient condition.

Definition 7.

A relator $\mathcal{R}$ is calledtopologically transitiveif for every $R,S\in \mathcal{R}$ there exists $T\in \mathcal{R}$ such that

$$T\subseteq R\circ S.$$

Topological transitivity is one of the standard structural properties considered for relators [4,9].

Proposition 3.

Suppose that $\mathcal{R}$ is reflexive and topologically transitive and that ${\tau }_{\mathcal{R}}$ is a topology. Then the relator interior has the finite-intersection property

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

whenever the witnessing relations can be chosen compatibly in $\mathcal{R}$.

Proof. 

The inclusion

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

is immediate. For the reverse inclusion, let $x\in {int}_{\mathcal{R}}\left(A\right)\cap {int}_{\mathcal{R}}\left(B\right)$. Then there are $R,S\in \mathcal{R}$ such that

$$R\left(x\right)\subseteq A,S\left(x\right)\subseteq B.$$

If the relator contains a common refinement T satisfying

$$T\left(x\right)\subseteq R\left(x\right)\cap S\left(x\right),$$

then

$$T\left(x\right)\subseteq A\cap B,$$

and consequently

$$x\in {int}_{\mathcal{R}}(A\cap B).$$

Thus the conclusion follows whenever $\mathcal{R}$ has the stated common-refinement property. □

Remark 4.

The common-refinement condition in the proposition is stronger than topological transitivity alone. This distinction is important when seeking a complete characterization of the relators for which $hO(X,\mathcal{R})$ is always a topology.

6. When Does $hO(X,\mathcal{R})={\tau }_{\mathcal{R}}$?

We next investigate the second question.

Proposition 4.

For every relator space,

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

whenever the defining h-openness condition is compatible with the relator interior in the sense that

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

for every $A\in {\tau }_{\mathcal{R}}$ and every nonempty proper $V\in {\tau }_{\mathcal{R}}$.

Proof. 

This is exactly the defining condition for every member of ${\tau }_{\mathcal{R}}$ to belong to $hO(X,\mathcal{R})$. □

The equality problem can therefore be reduced to the converse inclusion.

Theorem 6.

Suppose that ${\tau }_{\mathcal{R}}$ is a topology and that every h-open set is relator-open. Then

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

Conversely, if

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

then every h-open subset is relator-open.

Proof. 

The second statement is immediate. For the first statement, by assumption

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

By the preceding proposition,

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

Therefore equality follows. □

This leads to the following intrinsic criterion.

Corollary 1.

Assume that every relator-open set is h-open. Then

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

if and only if

$$A\in hO(X,\mathcal{R})⟹A\subseteq {int}_{\mathcal{R}}\left(A\right)$$

for every $A\subseteq X$.

Proof. 

The implication on the right is exactly

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

The reverse inclusion follows from the assumption. □

Remark 5.

For Pervin-type relators generated by a topology, the relation between the original topology and the relator-generated family is particularly transparent. Generalized topologies can be recovered from families of Pervin preorder relations, as described by Rassias, Salih and Száz [5].

7. Strictly Finer h-Topologies

Suppose now that $hO(X,\mathcal{R})$ is a topology and ${\tau }_{\mathcal{R}}$ is also a topology.

Definition 8.

We say that $hO(X,\mathcal{R})$ isstrictly finerthan ${\tau }_{\mathcal{R}}$ if

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

The following criterion is immediate but useful.

Proposition 5.

Assume that

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

Then

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

if and only if there exists a subset $A\subseteq X$ such that

$$A\notin {\tau }_{\mathcal{R}}$$

and

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

for every nonempty proper $V\in {\tau }_{\mathcal{R}}$.

Proof. 

If the strict inclusion holds, choose

$$A\in hO(X,\mathcal{R})\setminus {\tau }_{\mathcal{R}}.$$

The defining condition of h-openness gives the second condition. Conversely, if such an A exists, then

$$A\in hO(X,\mathcal{R})\setminus {\tau }_{\mathcal{R}},$$

so the inclusion is strict. □

Theorem 7.

Suppose that ${\tau }_{\mathcal{R}}$ is a topology and that ${\tau }_{\mathcal{R}}\subseteq hO(X,\mathcal{R})$. If there exists a non-relator-open set A such that

$$A\subseteq \bigcap _{\begin{array}{c}V\in {\tau }_{\mathcal{R}}\\ ⌀\ne V\ne X\end{array}}{int}_{\mathcal{R}}(A\cup V),$$

(6.1)

then

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

Proof. 

Condition (6.1) says precisely that A is h-open. Since $A\notin {\tau }_{\mathcal{R}}$, it belongs to

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

Hence the inclusion is strict. □

Example 1.

Let

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

and let

$${\tau }_{\mathcal{R}}=\{⌀,X,\left\{a\right\}\}.$$

Suppose that the relator is chosen so that

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

and

$$\{a,b\}\subseteq {int}_{\mathcal{R}}(\{a,b\}\cup \left\{a\right\})={int}_{\mathcal{R}}\left(\{a,b\}\right).$$

If $\{a,b\}\notin {\tau }_{\mathcal{R}}$, then $\{a,b\}$ is h-open but not relator-open. Consequently,

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

provided that $hO(X,\mathcal{R})$ is a topology.

8. The Associated h-Topology

The preceding discussion motivates the following terminology.

Definition 9.

If $hO(X,\mathcal{R})$ is a topology, we call $(X,hO(X,\mathcal{R}\left)\right)$ theassociated h-topological relator space.

We write ${\tau }_{h,\mathcal{R}}=hO(X,\mathcal{R})$.

Proposition 6.

Suppose that $hO(X,\mathcal{R})$ is a topology. Then ${\tau }_{h,\mathcal{R}}$ is an interior topology in the ordinary sense, and the operator

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

is an interior operator.

Proof. 

Because $hO(X,\mathcal{R})$ is a topology, the union of all its members contained in A is again h-open. Therefore ${Int}_{h,\mathcal{R}}\left(A\right)\subseteq A$,

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

$${Int}_{h,\mathcal{R}}\left({Int}_{h,\mathcal{R}}\left(A\right)\right)={Int}_{h,\mathcal{R}}\left(A\right),$$

and ${Int}_{h,\mathcal{R}}(A\cap B)={Int}_{h,\mathcal{R}}\left(A\right)\cap {Int}_{h,\mathcal{R}}\left(B\right)$. These are precisely the Kuratowski interior axioms. □

Remark 6.

We use the notation ${Int}_{h,\mathcal{R}}\left(A\right)$ for the h-interior of A with respect to the relator $\mathcal{R}$. This notation is chosen to avoid ambiguity and to prevent double-subscript errors in LaTeX.

9. h-Separation Axioms for Relator Spaces

We now develop the fourth question directly for relator spaces. Throughout this section assume that $hO(X,\mathcal{R})$ is a topology.

Definition 10.

The relator space $(X,\mathcal{R})$ is calledh-$T_0$if for every pair of distinct points $x,y\in X$, there exists $U\in hO(X,\mathcal{R})$ such that exactly one of $x,y$ belongs to U.

Definition 11.

The relator space $(X,\mathcal{R})$ is calledh-$T_1$if for every pair of distinct points $x,y\in X$, there exist $U,V\in hO(X,\mathcal{R})$ such that $x\in U$, $y\notin U$, and $y\in V$, $x\notin V$.

Definition 12.

The relator space $(X,\mathcal{R})$ is calledh-$T_2$if for every pair of distinct points $x,y\in X$, there exist $U,V\in hO(X,\mathcal{R})$ such that $x\in U$, $y\in V$, $U\cap V=⌀$.

These definitions are exactly the classical separation axioms applied to the associated topology

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

This associated-topology viewpoint has also been used in the recent study of h-separation properties [3].

Theorem 8.

Assume that $hO(X,\mathcal{R})$ is a topology. Then the following are equivalent:

(i)

$(X,\mathcal{R})$ is h-$T_0$;

(ii)

$(X,{\tau }_{h,\mathcal{R}})$ is $T_0$;

(iii)

for every distinct $x,y\in X$, there is an h-open set containing one of $x,y$ and not the other.

Proof. 

The equivalence follows directly from the definition $•{\tau }_{h,\mathcal{R}}=hO(X,\mathcal{R})$. □

Theorem 9.

Assume that $hO(X,\mathcal{R})$ is a topology. Then $(X,\mathcal{R})$ is h-$T_1$ if and only if every singleton $\left\{x\right\}$ is h-closed.

Proof. 

This is the standard singleton characterization of the $T_1$ axiom applied to the topology ${\tau }_{h,\mathcal{R}}$. □

Theorem 10.

Assume that $hO(X,\mathcal{R})$ is a topology. Then $(X,\mathcal{R})$ is h-$T_2$ if and only if for every distinct $x,y\in X$ there exist h-open sets $U,V$ satisfying

$$x\in U,y\in V,U\cap V=⌀.$$

Proof. 

This is precisely the definition of Hausdorffness for $(X,{\tau }_{h,\mathcal{R}})$. □

10. h-$R_0$ and h-$R_1$

We next formulate the corresponding regularity-type separation axioms.

Definition 13.

The relator space $(X,\mathcal{R})$ is calledh-$R_0$if for every $x\in X$ and every $U\in hO(X,\mathcal{R})$ with $x\in U$, ${Cl}_{h,\mathcal{R}}\left(\left\{x\right\}\right)\subseteq U$.

Here

$${Cl}_{h,\mathcal{R}}\left(A\right)=\bigcap \{F\subseteq X:F\mathrm{is}h-\mathrm{closed}\mathrm{and}A\subseteq F\}.$$

Proposition 7.

Assume that $hO(X,\mathcal{R})$ is a topology. Then $(X,\mathcal{R})$ is h-$R_0$ if and only if $(X,{\tau }_{h,\mathcal{R}})$ is an $R_0$ space.

Proof. 

Both statements assert that whenever x belongs to an open set U, the closure of $\left\{x\right\}$ is contained in U, where openness and closure are taken with respect to ${\tau }_{h,\mathcal{R}}$. □

Definition 14.

The relator space $(X,\mathcal{R})$ is calledh-$R_1$if for every pair of distinct points $x,y\in X$, whenever $y\notin {Cl}_{h,\mathcal{R}}\left(\left\{x\right\}\right)$, there exist h-open sets $U,V$ such that

$$x\in U,y\in V,U\cap V=⌀.$$

Proposition 8.

If $hO(X,\mathcal{R})$ is a topology, then $(X,\mathcal{R})$ is h-$R_1$ if and only if $(X,{\tau }_{h,\mathcal{R}})$ is $R_1$.

Proof. 

The statement follows immediately from the definition of ${\tau }_{h,\mathcal{R}}$ and the ordinary definition of an $R_1$ space. □

11. h-Regularity

Definition 15.

Assume that $hO(X,\mathcal{R})$ is a topology. We call $(X,\mathcal{R})$h-regularif for every point $x\in X$ and every h-closed set F with $x\notin F$, there exist $U,V\in hO(X,\mathcal{R})$ such that

$$x\in U,F\subseteq V,U\cap V=⌀.$$

Proposition 9.

The relator space $(X,\mathcal{R})$ is h-regular if and only if the associated topological space $(X,{\tau }_{h,\mathcal{R}})$ is regular.

Proof. 

The h-closed sets are precisely the closed sets of ${\tau }_{h,\mathcal{R}}$. Hence the assertion is exactly the ordinary definition of regularity in the associated topology. □

Remark 7.

This construction separates two levels of structure. The original relator $\mathcal{R}$ determines the relator-open family ${\tau }_{\mathcal{R}}$, while the h-open family determines the associated topology ${\tau }_{h,\mathcal{R}}$ whenever it is topological. Consequently, an h-separation axiom need not coincide with the corresponding separation axiom of ${\tau }_{\mathcal{R}}$.

12. Specialization Preorder

The separation axioms can also be described by the specialization preorder of the associated h-topology.

Definition 16.

Assume that $hO(X,\mathcal{R})$ is a topology. Define a relation ${⪯}_h$ on X by

$$x{⪯}_{h}y⟺x\in {Cl}_{h,\mathcal{R}}\left(\left\{y\right\}\right).$$

Equivalently,

$$x{⪯}_{h}y$$

if every h-open neighborhood of x contains y.

Proposition 10.

The relation ${⪯}_h$ is a preorder.

Proof. 

Reflexivity follows from $x\in {Cl}_{h,\mathcal{R}}\left(\left\{x\right\}\right)$. For transitivity, suppose $x{⪯}_{h}y\mathrm{and}y{⪯}_{h}z$. Every h-open neighborhood of x contains y, and every h-open neighborhood of y contains z. Hence every h-open neighborhood of x contains z, giving

$$x{⪯}_{h}z.$$

□

Theorem 11.

Assume that $hO(X,\mathcal{R})$ is a topology. Then $(X,\mathcal{R})$ is h-$T_0$ if and only if ${⪯}_h$ is antisymmetric.

Proof. 

This is the standard characterization of $T_0$ spaces by the specialization preorder applied to $(X,{\tau }_{h,\mathcal{R}})$. □

Theorem 12.

Assume that $hO(X,\mathcal{R})$ is a topology. If $(X,\mathcal{R})$ is h-$R_0$, then

$$x{⪯}_{h}y⟹y{⪯}_{h}x.$$

Consequently, ${⪯}_h$ is an equivalence relation.

Proof. 

This is the standard characterization of $R_0$ spaces by the symmetry of the specialization preorder. □

13. A Hierarchy of the Four Problems

The four questions raised in the introduction are related but logically distinct.

Proposition 11.

Suppose that $hO(X,\mathcal{R})$ is a topology and that ${\tau }_{\mathcal{R}}\subseteq hO(X,\mathcal{R})$. Then exactly one of the following two alternatives holds:

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

or

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

Proof. 

This is immediate from the inclusion ${\tau }_{\mathcal{R}}\subseteq hO(X,\mathcal{R})$. □

The important issue is therefore to determine when additional h-open sets occur.

Theorem 13.

Suppose that ${\tau }_{\mathcal{R}}$ and $hO(X,\mathcal{R})$ are topologies and ${\tau }_{\mathcal{R}}\subseteq hO(X,\mathcal{R})$. Then the following are equivalent:

(i)

$hO(X,\mathcal{R})={\tau }_{\mathcal{R}}$;

(ii)

every h-open set is relator-open;

(iii)

for every $A\subseteq X$,

$$A\subseteq \bigcap _{\begin{array}{c}V\in {\tau }_{\mathcal{R}}\\ ⌀\ne V\ne X\end{array}}{int}_{\mathcal{R}}(A\cup V)⟹A\subseteq {int}_{\mathcal{R}}\left(A\right).$$

Proof. 

The equivalence of (i) and (ii) is immediate. Assume (ii). If

$$A\subseteq \bigcap _{\begin{array}{c}V\in {\tau }_{\mathcal{R}}\\ ⌀\ne V\ne X\end{array}}{int}_{\mathcal{R}}(A\cup V),$$

then A is h-open. Hence, by (ii),

$$A\subseteq {int}_{\mathcal{R}}\left(A\right).$$

Conversely, assume (iii). If A is h-open, then it satisfies the displayed intersection condition. Hence

$$A\subseteq {int}_{\mathcal{R}}\left(A\right),$$

so $A\in {\tau }_{\mathcal{R}}$. Thus

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

Together with ${\tau }_{\mathcal{R}}\subseteq hO(X,\mathcal{R})$, this proves (i). □

14. Examples of the Two Extremes

Example 2

(Equality). Suppose that every subset which satisfies $A\subseteq {int}_{\mathcal{R}}(A\cup V)$ for every nonempty proper $V\in \tau R$ also satisfies $A\subseteq {int}_{\mathcal{R}}\left(A\right)$. Then $hO(X,\mathcal{R})={\tau }_{\mathcal{R}}$. Thus the h-construction produces no new open sets.

Example 3

(Strict enlargement). Suppose that there exists $A\subseteq X$ such that $A\notin \tau R$ but $A\subseteq {int}_{\mathcal{R}}(A\cup V)$ for every nonempty proper $V\in \tau R$. Then $A\in hO(X,\mathcal{R})\setminus \tau R$. Consequently, whenever $hO(X,\mathcal{R})$ is a topology, $\tau R⊊hO(X,\mathcal{R})$.

15. Relation with Classical Topological h-Open Sets

Suppose that $\tau$ is an ordinary topology on X. A classical way to obtain a relator associated with $\tau$ is through Pervin-type preorder relations. For $V\in \tau$, put

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

The family

$${\mathcal{R}}_{\tau }=\{R_V:V\in \tau \}$$

is the Pervin relator associated with $\tau$. Such constructions play an important role in representing generalized topologies by relators [5].

Proposition 12.

Let τ be a topology and let ${\mathcal{R}}_{\tau }$ be the associated Pervin relator. Then the relator framework provides a relational representation of the corresponding generalized topological structure.

Proof. 

This follows from the Pervin construction: the relations $R_V$ encode the members V of the topology, and the associated relator interior recovers the corresponding relational notion of openness. See [5] for the general construction. □

In this setting, the present theory can be compared with the ordinary h-open theory of topological spaces. In particular, when the relational construction reproduces $\tau$, the condition $A\subseteq {int}_{\mathcal{R}}(A\cup V)$ becomes the familiar condition $A\subseteq {Int}_{\tau }(A\cup V)$. Thus the relator theory extends the ordinary theory rather than replacing it.

16. Conclusions

We have developed a framework for studying h-open sets directly inside relator spaces. The construction starts from the relator interior

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

and defines h-openness by the stability condition

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

for all nonempty proper relator-open sets V. The topology problem is controlled by closure of the family of h-open sets under arbitrary unions and finite intersections. Condition (4.1) gives a useful sufficient criterion for finite intersection stability. The equality problem

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

is equivalent, under the natural inclusion, to the assertion that every h-open set is already relator-open. Conversely, the existence of a non-relator-open set satisfying the h-openness condition gives a direct criterion for strict refinement. Finally, whenever $hO(X,\mathcal{R})$ is a topology, it produces an associated topological relator space

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

This makes it possible to define h-$T_0$, h-$T_1$, h-$T_2$, h-$R_0$, h-$R_1$ and h-regularity simply as the corresponding separation properties of the associated topology.