Direct Continuity Properties of Relations in Relator Spaces

Traditionally, a function f of one topological space X ( T ) $$X(\mathcal {T})$$ to another Y ( S ) $$Y (\mathcal {S})$$ was called continuous if, under the notations T ( x ) = A ∈ T : x ∈ A and S ( y ) = B ∈ S : y ∈ B , $$\displaystyle \mathcal {T} (x)=\left \{ A\in \mathcal {T}: \ \ x\in A \right \} \qquad \mathrm {and}\qquad \mathcal {S} (y)=\left \{ B\in \mathcal {S}: \ \ y\in B \right \} , $$ the following property holds : (a) for each x ∈ X $$x\in X$$ and V ∈ S f ( x ) $$V\in \mathcal {S}\left (f (x)\right )$$ , there exists U ∈ T ( x ) $$U\in \mathcal {T} (x)$$ such that f [ U ] ⊆ V $$f [ U ]\subseteq V$$ , i.e., u ∈ U ⇒ f ( u ) ∈ V . $$\displaystyle u\in U \ \ \ \implies \ \ \ f (u)\in V . $$ This property, can, for instance, be reformulated in the following forms : (b) f is closure (nearness) preserving in the sense that, for all B ⊆ X $$B\subseteq X$$ , we have f cl T ( B ) ⊆ cl S f [ B ] $$f \left [ \operatorname {\mathrm {cl}}_{{ }_{\mathcal {T}}}(B) \right ]\subseteq \operatorname {\mathrm {cl}}_{{ }_{\mathcal {S}}}\left ( f [ B ] \right )$$ , i.e. , x ∈ cl T ( B ) ⇒ f ( x ) ∈ cl S f [ B ] . $$\displaystyle x\in \operatorname {\mathrm {cl}}_{{ }_{\mathcal {T}}}(B) \ \ \ \implies \ \ \ f (x)\in \operatorname {\mathrm {cl}}_{{ }_{\mathcal {S}}}\left ( f [ B ] \right ) . $$ (c) f is convergence preserving in the sense that, for any net ψ $$\psi $$ in X, we have f lim T ( ψ ) ⊆ lim S ( f ∘ ψ ) $$f \left [ \lim _{{ }_{\mathcal {T}}} (\psi ) \right ] \subseteq \lim _{{ }_{\mathcal {S}}}( f\circ \psi )$$ , i. e. , x ∈ lim T ( ψ ) ⇒ f ( x ) ∈ lim S ( f ∘ ψ ) . $$\displaystyle \textstyle x\in \lim _{{ }_{\mathcal {T}}}(\psi ) \ \ \ \implies \ \ \ f (x)\in \lim _{{ }_{\mathcal {S}}}( f\circ \psi ) . $$ However, most of the topologists prefer the following more elegant, but less convenient, reformulation : (d) the relation f −1 $$f^{-1}$$ open set preserving in the sense that V ∈ S ⇒ f −1 [ V ] ∈ T . $$\displaystyle V\in \mathcal {S} \ \ \ \implies \ \ \ f^{-1}[ V ]\in \mathcal {T} . $$ 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 ( R ) = ( X , R ) $$X(\mathcal {R})=(X, \mathcal {R})$$ consisting of a set X and a family R $$\mathcal {R}$$ of binary relations on X. Thus, relator spaces are immediate generalizations of ordered sets and uniform spaces. In the space X ( R ) $$X(\mathcal {R})$$ , for any x ∈ X $$x\in X$$ and A , B ⊆ X $$A, B\subseteq X$$ , we define A ∈ Int R ( B ) $$A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(B)$$ if R [ A ] ⊆ B $$R [ A ]\subseteq B$$ for some R ∈ R $$R\in \mathcal {R}$$ , and x ∈ int R ( A ) $$x\in \operatorname {\mathrm {int}}_{\mathcal {R}}(A)$$ if { x } ∈ Int R ( A ) $$\{x\}\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(A)$$ . And we define A ∈ τ R $$A\in \tau _{{ }_{\mathcal {R}}}$$ if A ∈ Int R ( A ) $$A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(A)$$ , A ∈ T R $$A\in \mathcal {T}_{\mathcal {R}}$$ if A ⊆ int R ( A ) $$A\subseteq \operatorname {\mathrm {int}}_{\mathcal {R}}(A)$$ , and A ∈ E R $$A\in \mathcal {E}_{\mathcal {R}}$$ if int R ( A ) ≠ ∅ $$ \operatorname {\mathrm {int}}_{\mathcal {R}}(A)\ne \emptyset $$ . Thus, having in mind continuity properties (d), (b), and (a), for any relation F on one relator space X ( R ) $$X(\mathcal {R})$$ to another Y ( S ) $$Y(\mathcal {S})$$ , we may also naturally say that : (1) F is proximal openness preserving if A ∈ τ R ⇒ F [ A ] ∈ τ S ; $$\displaystyle A\in \tau _{{ }_{\mathcal {R}}} \ \ \ \implies \ \ \ F [ A ]\in \tau _{{ }_{\mathcal {S}}} ; $$ (2) F is proximal interior preserving if, for all B ⊆ X $$B\subseteq X$$ , A ∈ Int R ( B ) ⇒ F [ A ] ∈ Int S F [ B ] . $$\displaystyle A\in \operatorname {\mathrm {Int}}_{\mathcal {R}}(B) \ \ \ \implies \ \ \ F [ A ]\in \operatorname {\mathrm {Int}}_{\mathcal {S}}\left ( F [ B ] \right ) . $$ (3) F is proximally mildly continuous if for each A ⊆ X $$A\subseteq X$$ and S ∈ S $$S\in \mathcal {S}$$ , there exists R ∈ R $$R\in \mathcal {R}$$ such that u ∈ R [ A ] ⇒ F ( u ) ∩ S [ F [ A ] ] ≠ ∅ . $$\displaystyle u\in R [ A ] \ \ \ \implies \ \ \ F (u)\cap S [ F [ A ] ]\ne \emptyset . $$ In a relator space X ( R ) $$X(\mathcal {R})$$ , for any two functions φ $$\varphi $$ and ψ $$\psi $$ of a relator space Γ ( U ) $$\varGamma (\mathcal {U})$$ to X, for instance, we may also naturally define ϕ ∈ Lim R ( ψ ) $$\phi \in \operatorname {\mathrm {Lim}}_{\mathcal {R}}(\psi )$$ if ( φ , ψ ) −1 [ R ] ∈ E U $$( \varphi , \psi )^{-1} [ R ]\in \mathcal {E}_{\mathcal {U}}$$ for all R ∈ R $$R\in \mathcal {R}$$ . Thus, having in mind continuity property (c), for any function f of one relator space X ( R ) $$X(\mathcal {R})$$ to another Y ( S ) $$Y (\mathcal {S})$$ , we may also say that (4) f is uniform convergence preserving if, for any relator space Γ ( U ) $$\varGamma (\mathcal {U})$$ and function ψ $$\psi $$ of Γ $$\varGamma $$ to X, φ ∈ Lim R ( ψ ) ⇒ f ∘ φ ∈ Lim R ( f ∘ ψ ) . $$\displaystyle \varphi \in \operatorname {\mathrm {Lim}}_{\mathcal {R}}(\psi ) \ \ \ \implies \ \ \ f\circ \varphi \in \operatorname {\mathrm {Lim}}_{\mathcal {R}}(f\circ \psi ) . $$ In the space X ( R ) $$X(\mathcal {R})$$ , for any A , B ⊆ X $$A, B\subseteq X$$ , we may also write A ∈ Lb R ( B ) $$A\in \operatorname {\mathrm {Lb}}_{\mathcal {R}}(B)$$ and B ∈ Ub R ( A ) $$B\in \operatorname {\mathrm {Ub}}_{\mathcal {R}}(A)$$ if A × B ⊆ R $$A \times B\subseteq R$$ for some R ∈ R $$R\in \mathcal {R}$$ . Moreover, we may also write Min R ( A ) = P ( A ) ∩ Lb R ( A ) $$ \operatorname {\mathrm {Min}}_{\mathcal {R}}(A)=\mathcal {P}(A)\cap \operatorname {\mathrm {Lb}}_{\mathcal {R}}(A)$$ and Sup R ( A ) = Min R Ub R ( A ) $$ \operatorname {\mathrm {Sup}}_{\mathcal {R}}(A)= \operatorname {\mathrm {Min}}_{\mathcal {R}} \left [ \operatorname {\mathrm {Ub}}_{\mathcal {R}}(A) \right ]$$ . 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 Lb R = Int R c ∘ C Y $$ \operatorname {\mathrm {Lb}}_{\mathcal {R}}= \operatorname {\mathrm {Int}}_{\mathcal {R}^{c}} \circ \mathcal {C}_{Y}$$ .