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

Themistocles M. Rassias (), Muwafaq M. Salih () and Árpád Száz ()
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Themistocles M. Rassias: National Technical University of Athens
Muwafaq M. Salih: College of Basic Education, University of Duhok
Árpád Száz: University of Debrecen

A chapter in Geometry and Non-Convex Optimization, 2025, pp 613-709 from Springer

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

Date: 2025
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DOI: 10.1007/978-3-031-87057-6_15

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