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, more generally, relators on X to Y may also be naturally considered. By using the following definitions, each minimal structure, generalized topology, or proper stack \(\mathcal {A}\) on X can be easily derived from the relator \(\mathcal {R}_{\mathcal {A}}\) consisting of all Pervin’s preorders \(R_{A}= A^{2}\cup (A^{c} \times X)\) with \(A\in \mathcal {A}\) . For any \(x\in X\) and \(A, B\subseteq X\) , we write (1) \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(B)\) if \(R [A] \subseteq B\) for some \(R\in \mathcal {R}\) ; (2) \(A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(B)\) if \(R [A] \cap B\ne \emptyset \) for all \(R\in \mathcal {R}\) ; (3) \(x\in \operatorname {\mathrm {int}}_{\mathcal {R}}(B)\) if \(\{x\}\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(B)\) ; (4) \(x\in \operatorname {\mathrm {cl}}_{\mathcal {R}}(B)\) if \(\{x\}\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(B)\) ; (5) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}\) if \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(A)\) ; (6) \(A\in \mathcal {T}_{\mathcal {R}}\) if \(A \subseteq \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\) ; (7) \(A\in \mathcal {E}_{\mathcal {R}}\) if \( \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\ne \emptyset \) ; (8) \(A\in \mathcal {N}_{\mathcal {R}}\) if \( \operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\notin \mathcal {E}_{\mathcal {R}}\) . In our former papers, motivated by some standard definitions in topological spaces, for a subset A of the relator space \(X(\mathcal {R})\) we, for instance, defined (a) \(A\in \mathcal {T}_{\mathcal {R}}^{s}\) if \(A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}\left (\operatorname {\mathrm {int}}_{\mathcal {R}}(A)\right)\) ; (b) \(A\in \mathcal {T}_{\mathcal {R}}^{p}\) if \(A\subseteq \operatorname {\mathrm {int}}_{\mathcal {R}}\left (\operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\right)\) ; (c) \(A\in \mathcal {T}_{\mathcal {R}}^{q}\) if \(V\subseteq A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(V)\) for some \(V\in \mathcal {T}_{\mathcal {R}}\) ; (d) \(A\in \mathcal {T}_{\mathcal {R}}^{ps}\) if \(A\subseteq V\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\) for some \(V\in \mathcal {T}_{\mathcal {R}}\) . Now, in addition to the above basic definitions, we shall also consider the following new definitions: (A) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{s}\) if \(A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}} \left [\operatorname {\mathrm {Int}}_{\mathcal {R}}(A) \right]\) ; (B) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{p}\) if \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}} \left [\operatorname {\mathrm {Cl}}_{\mathcal {R}}(A) \right]\) ; (C) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{ms}\) if \(A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}} \left [\operatorname {\mathrm {Int}}_{\mathcal {R}}(A) \right]\) ; (D) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{mp}\) if \(A\subseteq \operatorname {\mathrm {int}}_{\mathcal {R}} \left [\operatorname {\mathrm {Cl}}_{\mathcal {R}}(A) \right]\) ; (E) \(A\in \mathcal {T}_{\mathcal {R}}^{ms}\) if \(A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}} \left (\operatorname {\mathrm {int}}_{\mathcal {R}}(A) \right)\) ; (F) \(A\in \mathcal {T}_{\mathcal {R}}^{mp}\) if \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}} \left (\operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\right)\) ; (G) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{q}\) if \(V\subseteq A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(V)\) for some \(V\in \tau _{\scriptscriptstyle \mathcal {R}}\) ; (H) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{ps}\) if \(A\subseteq V\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\) for some \(V\in \tau _{\scriptscriptstyle \mathcal {R}}\) ; (I) \(A\in \mathcal {E}_{\mathcal {R}}^{q}\) if \(V\subseteq A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(V)\) for some \(V\in \mathcal {E}_{\mathcal {R}}\) ; (J) \(A\in \mathcal {E}_{\mathcal {R}}^{ps}\) if \(A\subseteq V\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\) for some \(V\in \mathcal {E}_{\mathcal {R}}\) ; (K) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{wq}\) if \(V\subseteq A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(V)\) for some \(V\in \tau _{\scriptscriptstyle \mathcal {R}}\) ; (L) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{wps}\) if \(A\subseteq V\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(A)\) for some \(V\in \tau _{\scriptscriptstyle \mathcal {R}}\) ; (M) \(A\in \mathcal {T}_{\mathcal {R}}^{wq}\) if \(V\subseteq A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(V)\) for some \(V\in \mathcal {T}_{\mathcal {R}}\) ; (N) \(A\in \mathcal {T}_{\mathcal {R}}^{wps}\) if \(A\subseteq V\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(A)\) for some \(V\in \mathcal {T}_{\mathcal {R}}\) ; (O) \(A\in \mathcal {E}_{\mathcal {R}}^{wq}\) if \(V\subseteq A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(V)\) for some \(V\in \mathcal {E}_{\mathcal {R}}\) ; (P) \(A\in \mathcal {E}_{\mathcal {R}}^{wps}\) if \(A\subseteq V\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(A)\) for some \(V\in \mathcal {E}_{\mathcal {R}}\) . And, analogously to our former papers, we shall establish some characterizations and set theoretic properties of the families \(\tau _{\scriptscriptstyle \mathcal {R}}^{\kappa}\) , \(\mathcal {T}_{\mathcal {R}}^{\kappa}\) and \(\mathcal {E}_{\mathcal {R}}^{\kappa}\) with the operations \(\kappa \) considered in definitions (A)–(P).

Characterizations and Set Theoretic Properties of Some Generalized Open and Fat Sets in Relator Spaces

章节
发表日期：

2025年

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Themistocles M. Rassias,
Muwafaq M. Salih,
Árpád Száz

摘要

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, more generally, relators on X to Y may also be naturally considered. By using the following definitions, each minimal structure, generalized topology, or proper stack \(\mathcal {A}\) on X can be easily derived from the relator \(\mathcal {R}_{\mathcal {A}}\) consisting of all Pervin’s preorders \(R_{A}= A^{2}\cup (A^{c} \times X)\) with \(A\in \mathcal {A}\) . For any \(x\in X\) and \(A, B\subseteq X\) , we write (1) \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(B)\) if \(R [A] \subseteq B\) for some \(R\in \mathcal {R}\) ; (2) \(A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(B)\) if \(R [A] \cap B\ne \emptyset \) for all \(R\in \mathcal {R}\) ; (3) \(x\in \operatorname {\mathrm {int}}_{\mathcal {R}}(B)\) if \(\{x\}\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(B)\) ; (4) \(x\in \operatorname {\mathrm {cl}}_{\mathcal {R}}(B)\) if \(\{x\}\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(B)\) ; (5) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}\) if \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(A)\) ; (6) \(A\in \mathcal {T}_{\mathcal {R}}\) if \(A \subseteq \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\) ; (7) \(A\in \mathcal {E}_{\mathcal {R}}\) if \( \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\ne \emptyset \) ; (8) \(A\in \mathcal {N}_{\mathcal {R}}\) if \( \operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\notin \mathcal {E}_{\mathcal {R}}\) . In our former papers, motivated by some standard definitions in topological spaces, for a subset A of the relator space \(X(\mathcal {R})\) we, for instance, defined (a) \(A\in \mathcal {T}_{\mathcal {R}}^{s}\) if \(A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}\left (\operatorname {\mathrm {int}}_{\mathcal {R}}(A)\right)\) ; (b) \(A\in \mathcal {T}_{\mathcal {R}}^{p}\) if \(A\subseteq \operatorname {\mathrm {int}}_{\mathcal {R}}\left (\operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\right)\) ; (c) \(A\in \mathcal {T}_{\mathcal {R}}^{q}\) if \(V\subseteq A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(V)\) for some \(V\in \mathcal {T}_{\mathcal {R}}\) ; (d) \(A\in \mathcal {T}_{\mathcal {R}}^{ps}\) if \(A\subseteq V\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\) for some \(V\in \mathcal {T}_{\mathcal {R}}\) . Now, in addition to the above basic definitions, we shall also consider the following new definitions: (A) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{s}\) if \(A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}} \left [\operatorname {\mathrm {Int}}_{\mathcal {R}}(A) \right]\) ; (B) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{p}\) if \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}} \left [\operatorname {\mathrm {Cl}}_{\mathcal {R}}(A) \right]\) ; (C) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{ms}\) if \(A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}} \left [\operatorname {\mathrm {Int}}_{\mathcal {R}}(A) \right]\) ; (D) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{mp}\) if \(A\subseteq \operatorname {\mathrm {int}}_{\mathcal {R}} \left [\operatorname {\mathrm {Cl}}_{\mathcal {R}}(A) \right]\) ; (E) \(A\in \mathcal {T}_{\mathcal {R}}^{ms}\) if \(A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}} \left (\operatorname {\mathrm {int}}_{\mathcal {R}}(A) \right)\) ; (F) \(A\in \mathcal {T}_{\mathcal {R}}^{mp}\) if \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}} \left (\operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\right)\) ; (G) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{q}\) if \(V\subseteq A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(V)\) for some \(V\in \tau _{\scriptscriptstyle \mathcal {R}}\) ; (H) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{ps}\) if \(A\subseteq V\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\) for some \(V\in \tau _{\scriptscriptstyle \mathcal {R}}\) ; (I) \(A\in \mathcal {E}_{\mathcal {R}}^{q}\) if \(V\subseteq A\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(V)\) for some \(V\in \mathcal {E}_{\mathcal {R}}\) ; (J) \(A\in \mathcal {E}_{\mathcal {R}}^{ps}\) if \(A\subseteq V\subseteq \operatorname {\mathrm {cl}}_{\mathcal {R}}(A)\) for some \(V\in \mathcal {E}_{\mathcal {R}}\) ; (K) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{wq}\) if \(V\subseteq A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(V)\) for some \(V\in \tau _{\scriptscriptstyle \mathcal {R}}\) ; (L) \(A\in \tau _{\scriptscriptstyle \mathcal {R}}^{wps}\) if \(A\subseteq V\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(A)\) for some \(V\in \tau _{\scriptscriptstyle \mathcal {R}}\) ; (M) \(A\in \mathcal {T}_{\mathcal {R}}^{wq}\) if \(V\subseteq A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(V)\) for some \(V\in \mathcal {T}_{\mathcal {R}}\) ; (N) \(A\in \mathcal {T}_{\mathcal {R}}^{wps}\) if \(A\subseteq V\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(A)\) for some \(V\in \mathcal {T}_{\mathcal {R}}\) ; (O) \(A\in \mathcal {E}_{\mathcal {R}}^{wq}\) if \(V\subseteq A\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(V)\) for some \(V\in \mathcal {E}_{\mathcal {R}}\) ; (P) \(A\in \mathcal {E}_{\mathcal {R}}^{wps}\) if \(A\subseteq V\in \operatorname {\mathrm {Cl}}_{\mathcal {R}}(A)\) for some \(V\in \mathcal {E}_{\mathcal {R}}\) . And, analogously to our former papers, we shall establish some characterizations and set theoretic properties of the families \(\tau _{\scriptscriptstyle \mathcal {R}}^{\kappa}\) , \(\mathcal {T}_{\mathcal {R}}^{\kappa}\) and \(\mathcal {E}_{\mathcal {R}}^{\kappa}\) with the operations \(\kappa \) considered in definitions (A)–(P).