Traditionally, a function f of one topological space \(X(\mathcal {T})\) to another \(Y (\mathcal {S})\) was called continuous if, under the notations \(\displaystyle \mathcal {T} (x)=\left \{ A\in \mathcal {T}: \ \ x\in A \right \} \qquad \mathrm {and}\qquad \mathcal {S} (y)=\left \{ B\in \mathcal {S}: \ \ y\in B \right \} , \) the following property holds : \(f [ U ]\subseteq V\) , i.e., \(\displaystyle u\in U \ \ \ \implies \ \ \ f (u)\in V . \) This property, can, for instance, be reformulated in the following forms : have \(f \left [ \lim _{{ }_{\mathcal {T}}} (\psi ) \right ] \subseteq \lim _{{ }_{\mathcal {S}}}( f\circ \psi )\) , i. e. , \(\displaystyle \textstyle x\in \lim _{{ }_{\mathcal {T}}}(\psi ) \ \ \ \implies \ \ \ f (x)\in \lim _{{ }_{\mathcal {S}}}( f\circ \psi ) . \) However, most of the topologists prefer the following more elegant, but less convenient, reformulation : The above continuity properties have been intensively investigated by a great number of mathematicians not only in generalized topological spaces but also in generalized neighborhood, closure, and convergence spaces. Therefore, it seems reasonable to treat these continuity properties also in relator spaces developed by the second author and his collaborators. Namely, they provide the most convenient framework for such continuity considerations too. A relator space, in a narrower sense, is an ordered pair \(X(\mathcal {R})=(X, \mathcal {R})\) consisting of a set X and a family \(\mathcal {R}\) of binary relations on X. Thus, relator spaces are immediate generalizations of ordered sets and uniform spaces. In the space \(X(\mathcal {R})\) , for any \(x\in X\) and \(A, B\subseteq X\) , we define \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(B)\) if \(R [ A ]\subseteq B\) for some \(R\in \mathcal {R}\) , and \(x\in \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\) if \(\{x\}\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(A)\) . And we define \(A\in \tau _{{ }_{\mathcal {R}}}\) if \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(A)\) , \(A\in \mathcal {T}_{\mathcal {R}}\) if \(A\subseteq \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\) , and \(A\in \mathcal {E}_{\mathcal {R}}\) if \( \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\ne \emptyset \) . Thus, having in mind continuity properties (d), (b), and (a), for any relation F on one relator space \(X(\mathcal {R})\) to another \(Y(\mathcal {S})\) , we may also naturally say that : In a relator space \(X(\mathcal {R})\) , for any two functions \(\varphi \) and \(\psi \) of a relator space \(\varGamma (\mathcal {U})\) to X, for instance, we may also naturally define \(\phi \in \operatorname {\mathrm {Lim}}_{\mathcal {R}}(\psi )\) if \(( \varphi , \psi )^{-1} [ R ]\in \mathcal {E}_{\mathcal {U}}\) for all \(R\in \mathcal {R}\) . Thus, having in mind continuity property (c), for any function f of one relator space \(X(\mathcal {R})\) to another \(Y (\mathcal {S})\) , we may also say that In the space \(X(\mathcal {R})\) , for any \(A, B\subseteq X\) , we may also write \(A\in \operatorname {\mathrm {Lb}}_{\mathcal {R}}(B)\) and \(B\in \operatorname {\mathrm {Ub}}_{\mathcal {R}}(A)\) if \(A \times B\subseteq R\) for some \(R\in \mathcal {R}\) . Moreover, we may also write \( \operatorname {\mathrm {Min}}_{\mathcal {R}}(A)=\mathcal {P}(A)\cap \operatorname {\mathrm {Lb}}_{\mathcal {R}}(A)\) and \( \operatorname {\mathrm {Sup}}_{\mathcal {R}}(A)= \operatorname {\mathrm {Min}}_{\mathcal {R}} \left [ \operatorname {\mathrm {Ub}}_{\mathcal {R}}(A) \right ]\) . Thus, in addition to continuity property (2), several other similar preservation properties can also be investigated. However, these are not independent of property (2). Since, for instance, it can be shown that \( \operatorname {\mathrm {Lb}}_{\mathcal {R}}= \operatorname {\mathrm {Int}}_{\mathcal {R}^{c}} \circ \mathcal {C}_{Y}\) .

Direct Continuity Properties of Relations in Relator Spaces

章节
发表日期：

2025年

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Themistocles M. Rassias,
Árpád Száz

摘要

Traditionally, a function f of one topological space \(X(\mathcal {T})\) to another \(Y (\mathcal {S})\) was called continuous if, under the notations \(\displaystyle \mathcal {T} (x)=\left \{ A\in \mathcal {T}: \ \ x\in A \right \} \qquad \mathrm {and}\qquad \mathcal {S} (y)=\left \{ B\in \mathcal {S}: \ \ y\in B \right \} , \) the following property holds : \(f [ U ]\subseteq V\) , i.e., \(\displaystyle u\in U \ \ \ \implies \ \ \ f (u)\in V . \) This property, can, for instance, be reformulated in the following forms : have \(f \left [ \lim _{{ }_{\mathcal {T}}} (\psi ) \right ] \subseteq \lim _{{ }_{\mathcal {S}}}( f\circ \psi )\) , i. e. , \(\displaystyle \textstyle x\in \lim _{{ }_{\mathcal {T}}}(\psi ) \ \ \ \implies \ \ \ f (x)\in \lim _{{ }_{\mathcal {S}}}( f\circ \psi ) . \) However, most of the topologists prefer the following more elegant, but less convenient, reformulation : The above continuity properties have been intensively investigated by a great number of mathematicians not only in generalized topological spaces but also in generalized neighborhood, closure, and convergence spaces. Therefore, it seems reasonable to treat these continuity properties also in relator spaces developed by the second author and his collaborators. Namely, they provide the most convenient framework for such continuity considerations too. A relator space, in a narrower sense, is an ordered pair \(X(\mathcal {R})=(X, \mathcal {R})\) consisting of a set X and a family \(\mathcal {R}\) of binary relations on X. Thus, relator spaces are immediate generalizations of ordered sets and uniform spaces. In the space \(X(\mathcal {R})\) , for any \(x\in X\) and \(A, B\subseteq X\) , we define \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(B)\) if \(R [ A ]\subseteq B\) for some \(R\in \mathcal {R}\) , and \(x\in \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\) if \(\{x\}\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(A)\) . And we define \(A\in \tau _{{ }_{\mathcal {R}}}\) if \(A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(A)\) , \(A\in \mathcal {T}_{\mathcal {R}}\) if \(A\subseteq \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\) , and \(A\in \mathcal {E}_{\mathcal {R}}\) if \( \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\ne \emptyset \) . Thus, having in mind continuity properties (d), (b), and (a), for any relation F on one relator space \(X(\mathcal {R})\) to another \(Y(\mathcal {S})\) , we may also naturally say that : In a relator space \(X(\mathcal {R})\) , for any two functions \(\varphi \) and \(\psi \) of a relator space \(\varGamma (\mathcal {U})\) to X, for instance, we may also naturally define \(\phi \in \operatorname {\mathrm {Lim}}_{\mathcal {R}}(\psi )\) if \(( \varphi , \psi )^{-1} [ R ]\in \mathcal {E}_{\mathcal {U}}\) for all \(R\in \mathcal {R}\) . Thus, having in mind continuity property (c), for any function f of one relator space \(X(\mathcal {R})\) to another \(Y (\mathcal {S})\) , we may also say that In the space \(X(\mathcal {R})\) , for any \(A, B\subseteq X\) , we may also write \(A\in \operatorname {\mathrm {Lb}}_{\mathcal {R}}(B)\) and \(B\in \operatorname {\mathrm {Ub}}_{\mathcal {R}}(A)\) if \(A \times B\subseteq R\) for some \(R\in \mathcal {R}\) . Moreover, we may also write \( \operatorname {\mathrm {Min}}_{\mathcal {R}}(A)=\mathcal {P}(A)\cap \operatorname {\mathrm {Lb}}_{\mathcal {R}}(A)\) and \( \operatorname {\mathrm {Sup}}_{\mathcal {R}}(A)= \operatorname {\mathrm {Min}}_{\mathcal {R}} \left [ \operatorname {\mathrm {Ub}}_{\mathcal {R}}(A) \right ]\) . Thus, in addition to continuity property (2), several other similar preservation properties can also be investigated. However, these are not independent of property (2). Since, for instance, it can be shown that \( \operatorname {\mathrm {Lb}}_{\mathcal {R}}= \operatorname {\mathrm {Int}}_{\mathcal {R}^{c}} \circ \mathcal {C}_{Y}\) .