A General Framework for Studying Certain Generalized Topologically Open Sets in Relator Spaces

Abstract

A family $\mathcal {R}$ of binary relations on a set X is called a relator on X, and the ordered pair $X(\mathcal {R})=( X, \mathcal {R})$ is called a relator space. Sometimes relators on X to Y  are also considered.

By using an obvious definition of the generated open sets, each generalized topology $\mathcal {T}$ on X can be easily derived from the family of all Pervin’s preorder relations R _V = V ² ∪ V ^c  × X with $V\in \mathcal {T}$, where $V^{2}={}V\!\times \!V$ and V ^c = X ∖ V .

For a subset A of the relator space $X(\mathcal {R})$, we define

$$\displaystyle A^{\circ }= \operatorname {\mathrm {int}}_{ {\mathcal {R}}}(A)= \big \{x\in X: \ \ \ \exists \ R\in \mathcal {R}: \,\ \ R\,(x)\subseteq {}A\,\big \} $$

and . And, for instance, we write if A ⊆ A ^∘.

Moreover, following some basic definitions in topological spaces, for a subset A of the relator space $X(\mathcal {R})$ we write

The members of the above families will be called the topologically regular open, preopen, semi-open, α-open, β-open, a-open and b-open subsets of the relator space $X(\mathcal {R})$, respectively.

In our former papers, having in mind the original definitions of N. Levine [49] and H. H. Corson and E. Michael [11], we have also investigated four further, closely related, families of generalized topologically open sets in $X(\mathcal {R})$.

Now, we shall offer a general framework for studying these families. Moreover, motivated by a definition of Á. Császár [15] and his predecessors, we shall also consider a further important class of generalized topologically open sets.

For the latter purpose, for a subset A of the relator space $X(\mathcal {R})$, we shall write

• (8)  if  A ^{− ∘}⊆ A ^{∘ −}.

Thus, according to Császár’s terminology, the members of the family should be called the topologically quasi-open subsets of the relator space $X(\mathcal {R})$. However, in the earlier literature, these sets have been studied under different names.

While, for the former purpose, for any two subsets A and B of the relator space $X(\mathcal {R})$ we shall write

• (9) and  if  A ⊆ B ⊆ A ⁻.

Moreover, for a family $\mathcal {A}$ of subsets of $X(\mathcal {R})$ we shall define

• (10)  and  .

Thus, $\mathcal {A}^{\hskip 1mm\ell }$ and $\mathcal {A}^{{\hskip 1mm}u}$ may be called the lower and upper nearness closures of $\mathcal {A}$, respectively. Namely, if , then we may naturally say that A is near to B from below and B is near to A from above.

The most important particular cases are when $\mathcal {A}$ is a minimal structure or a generalized topology on X. Or even more specially, $\mathcal {A}$ is one of the families , or .