Set-Theoretic Properties of 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 ℛ 𝒯 $$\mathcal {R}_{\mathcal {T}}$$ of all Pervin’s preorder relations R V = V2 ∪ (Vc × X) with V ∈ 𝒯 $$V\in \mathcal {T}$$ , where V2 = V × V and Vc = 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 $$\displaystyle {A^{\circ }= \operatorname {\mathrm {int}}_{\mathcal {R}}\,(A)= \big \{x\in X: \ \quad \exists \,\ R\in \mathcal {R}: \quad R\,(x)\subseteq A\,\big \}} $$ and A − = cl ℛ ( A ) = int ℛ ( A c ) c $$A^{-}= \operatorname {\mathrm {cl}}_{\mathcal {R}}\,(A)= \operatorname {\mathrm {int}}_{\mathcal {R}}\,(A^{c})^{c}$$ . And, for instance, we also define 𝒯 ℛ = A ⊆ X : A ⊆ A ∘ and ℱ ℛ = A ⊆ X : A c ∈ 𝒯 ℛ . $$\displaystyle {\mathcal {T}_{\mathcal {R}}=\big \{A\subseteq X: \,\ \ A\subseteq A^{\circ }\,\big \} \ \qquad \mbox{and}\qquad \ \mathcal {F}_{\mathcal {R}}=\big \{A\subseteq X: \,\ \ A^{c}\in \mathcal {T}_{\mathcal {R}}\,\big \}.} $$ Moreover, motivated by some basic definitions in topological spaces, for a subset A of the relator space X ( ℛ ) $$X(\mathcal {R})$$ we shall write (1) A ∈ 𝒯 ℛ r $$A\in \mathcal {T}_{\mathcal {R}}^{r}$$ if A = A−∘ ; (2) A ∈ 𝒯 ℛ p $$A\in \mathcal {T}_{\mathcal {R}}^{p}$$ if A ⊆ A−∘ ; (3) A ∈ 𝒯 ℛ s $$A\in \mathcal {T}_{\mathcal {R}}^{s}$$ if A ⊆ A∘− ; (4) A ∈ 𝒯 ℛ α $$A\in \mathcal {T}_{\mathcal {R}}^{\alpha }$$ if A ⊆ A∘−∘ ; (5) A ∈ 𝒯 ℛ β $$A\in \mathcal {T}_{\mathcal {R}}^{\beta }$$ if A ⊆ A− ∘ − ; (6) A ∈ 𝒯 ℛ a $$A\in \mathcal {T}_{\mathcal {R}}^{a}$$ if A ⊆ A−∘∩ A∘−; (7) A ∈ 𝒯 ℛ b $$A\in \mathcal {T}_{\mathcal {R}}^{b}$$ if A ⊆ A−∘∪ A∘−; (8) A ∈ 𝒯 ℛ q $$A\in \mathcal {T}_{\mathcal {R}}^{q}$$ if there exists V ∈ 𝒯 ℛ $$V\in \mathcal {T}_{\mathcal {R}}\ $$ such that V ⊆ A ⊆ V−; (9) A ∈ 𝒯 ℛ ps $$A\in \mathcal {T}_{\mathcal {R}}^{ps}$$ if there exists V ∈ 𝒯 ℛ $$V\in \mathcal {T}_{\mathcal {R}}\ $$ such that A ⊆ V ⊆ A−; (10) A ∈ 𝒯 ℛ γ $$A\in \mathcal {T}_{\mathcal {R}}^{\gamma }$$ if there exists V ∈ 𝒯 ℛ s $$V\in \mathcal {T}_{\mathcal {R}}^{s}\ $$ such that A ⊆ V ⊆ A− ; (11) A ∈ 𝒯 ℛ δ $$A\in \mathcal {T}_{\mathcal {R}}^{\delta }$$ if there exists V ∈ 𝒯 ℛ p $$V\in \mathcal {T}_{\mathcal {R}}^{p}\ $$ such that V ⊆ A ⊆ V−. And, the members of the above families will be called the topologically regular open, preopen, semi-open, α-open, β-open, a-open, b-open, quasi-open, pseudo-open, γ-open, and δ-open subsets of the relator space X ( ℛ ) $$X(\mathcal {R})$$ , respectively. In a former paper, we have systematically investigated the various relationships among the families 𝒯 ℛ κ $$\mathcal {T}_{\mathcal {R}}^{\kappa }$$ . Moreover, we have tried to establish several illuminating characterizations of the families 𝒯 ℛ κ $$\mathcal {T}_{\mathcal {R}}^{\kappa }$$ . Here, we shall mainly be interested in the most simple set-theoretic properties of the families 𝒯 ℛ κ $$\mathcal {T}_{\mathcal {R}}^{\kappa }$$ . First of all, we shall briefly investigate their dual families ℱ ℛ κ = { A ⊆ X : A c ∈ 𝒯 ℛ κ } $$\mathcal {F}_{\mathcal {R}}^{\kappa }=\{A\subseteq X: \ \ A^{c}\in \mathcal {T}_{\mathcal {R}}^{\kappa }\}$$ . Then, we shall establish some intrinsic characterizations of the families 𝒯 ℛ κ $$\mathcal {T}_{\mathcal {R}}^{\kappa }$$ . Moreover, we shall give some necessary and sufficient conditions in order that ∅, {x}, with x ∈ X, and X could be contained in 𝒯 ℛ κ $$\mathcal {T}_{\mathcal {R}}^{\kappa }$$ . Finally, we shall show that, with the exception of 𝒯 ℛ r $$\mathcal {T}_{\mathcal {R}}^{r}$$ , the families 𝒯 ℛ κ $$\mathcal {T}_{\mathcal {R}}^{\kappa }$$ are closed under arbitrary unions. Moreover, for every 𝒯 ℛ κ $$\mathcal {T}_{\mathcal {R}}^{\kappa }$$ , we shall try to determine those subsets A of X which satisfy A ∩ B ∈ 𝒯 ℛ κ $$A\cap B\in \mathcal {T}_{\mathcal {R}}^{\kappa }$$ for all B ∈ 𝒯 ℛ κ $$B\in \mathcal {T}_{\mathcal {R}}^{\kappa }$$ . Furthermore, we shall indicate that, analogously to the family 𝒯 ℛ $$\mathcal {T}_{\mathcal {R}}$$ of all topologically open subsets of the relator spaces X ( ℛ ) $$X(\mathcal {R})$$ , the families 𝒯 ℛ κ $$\mathcal {T}_{\mathcal {R}}^{\kappa }$$ can also be used to introduce some interesting classifications of relators.