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

A family R $$\mathcal {R}$$ of binary relations on a set X is called a relator on X, and the ordered pair X ( R ) = ( X , R ) $$ X(\mathcal {R})=( X, \mathcal {R})$$ 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 T $$\mathcal {T}$$ on X can be easily derived from the family of all Pervin’s preorder relations R V = V 2 ∪ V c × X with V ∈ T $$V\in \mathcal {T}$$ , where V 2 = V × V $$V^{2}={}V\!\times \!V$$ and V c = X ∖ V . For a subset A of the relator space X ( R ) $$X(\mathcal {R})$$ , we define A ∘ = int R ( A ) = { x ∈ X : ∃ R ∈ R : R ( x ) ⊆ A } $$\displaystyle A^{\circ }= \operatorname {\mathrm {int}}_{ {\mathcal {R}}}(A)= \big \{x\in X: \ \ \ \exists \ R\in \mathcal {R}: \,\ \ R\,(x)\subseteq {}A\,\big \} $$ 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 ( R ) $$X(\mathcal {R})$$ 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 ( R ) $$X(\mathcal {R})$$ , 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 ( R ) $$X(\mathcal {R})$$ . 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 ( R ) $$X(\mathcal {R})$$ , 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 ( R ) $$X(\mathcal {R})$$ . 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 ( R ) $$X(\mathcal {R})$$ we shall write (9) and if A ⊆ B ⊆ A −. Moreover, for a family A $$\mathcal {A}$$ of subsets of X ( R ) $$X(\mathcal {R})$$ we shall define (10) and . Thus, A ℓ $$\mathcal {A}^{\hskip 1mm\ell }$$ and A u $$\mathcal {A}^{{\hskip 1mm}u}$$ may be called the lower and upper nearness closures of A $$\mathcal {A}$$ , 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 A $$\mathcal {A}$$ is a minimal structure or a generalized topology on X. Or even more specially, A $$\mathcal {A}$$ is one of the families , or .