Multi-valued continuous mappings of figure spaces (Q2508791)

Let \(A\) and \(B\) be two nonempty sets and binary relations \(s\subset A\times B\) and \(f_s: A\to 2^B\) from the set \(A\) into the Boolean \(2^B\) by the the rule \(f_s(a)= \{b\in B:(a,b)\in s\}\) for each \(a\in A\) and call an \(s\)-mapping the mapping \(s(a)= f_s(a)\), An \(m\)-mapping is an \(s\)-mapping from \(A\) into \(B\) such that \(m(a)\neq\emptyset\) for each \(a\in A\). Let \({\mathcal A}\subset 2^X\) and \({\mathcal B}\subset 2^Y\) be nonempty pairs of the Booleans \(2^X\) and \(2^Y\), respectively, i.e. \({\mathcal A}\) and \({\mathcal B}\) are systems of sets. We consider that -- the system \({\mathcal A}\) is covered by the system \({\mathcal B}\) and the system \({\mathcal B}\) covers the system \({\mathcal A}\), i.e. is cofinal for the system \({\mathcal A}\), and write \({\mathcal A}\subseteq_*{\mathcal B}\) if \(\forall A\in{\mathcal A}\), \(\exists B\subset{\mathcal B}\) \((A\subseteq B)\). -- \({\mathcal A}\) is a support system for the system \({\mathcal B}\), i.e. is coinitial, and we write \({\mathcal B}\subseteq^{-1}_*{\mathcal A}\) if \(\forall B\in{\mathcal B}\), \(\exists A\in{\mathcal A}\) \((A\subseteq B)\). For an \(s\)-mapping from \(X\) into \(Y\), the image of the system \({\mathcal A}\) is the system of images of all sets \(A\in{\mathcal A}\), which is denoted by \(s({\mathcal A})\), and the preimage (\(o\)-preimage) of the system \({\mathcal B}\) is the system of preimages (\(o\)-preimages) of all sets \(B\in{\mathcal B}\), which is denoted by \(s^{-1}({\mathcal B})\) (according, \(s^{-o}({\mathcal B})\)). If a system of sets \({\mathcal F}\subseteq 2^X\) covers the set \(X\), then we call the pair \((X,{\mathcal F})\) a figure space \((X,{\mathcal F})\) [\textit{V. Ya. Burdyuk}, Cybern. Syst. Anal. 40, No. 2, 169-183 (2004); translation from Kibern. Sist. Anal. 2004, No. 2, 24-44 (2004; Zbl 1097.54002)], call the sets \(F\in{\mathcal F}\) figures, and the set \(X\) the spanning set of the figure space. For an arbitrary point \(x\in X\) of the figure space \((X,{\mathcal F})\), we denote by \(Cx\) the system of all the figures \(F\in{\mathcal F}\) each of which covers the point \(x\), i.e. we have \(Cx= \{F\in{\mathcal F}: x\in F\}\). For arbitrary figure spaces \((X,{\mathcal F})\) and \((Y,{\mathcal G})\) and an arbitrary \(m\)-mapping from \(X\) into \(Y\), we write \(m:(x,{\mathcal F})\to (Y,{\mathcal G})\), call it an \(m\)-mapping of figure spaces, and also introduce the following notations: \(Cmx\) is the system of all the figures \(G\in{\mathcal G}\) each of which covers the image of a point \(x\), i.e., we have \(Cmx= \{G\in{\mathcal G}: m(x)\subseteq G\}\). \(Tmx\) is the system of all the figures \(G\in{\mathcal G}\) each of which is adjacent to the image of a point \(x\), i.e. we have \(Tmx= \{G\in{\mathcal G}: m(x)\cap G\neq\emptyset\}\). Let \(m:(X,{\mathcal F})\to (Y,{\mathcal G})\) be an arbitrary \(m\)-mapping of figure spaces. We form the following eight monadic predicates \(P_j(x)\) of an object variable \(x\in X\): \(P_1(x)= m^{-0}(Tmx)- \{\emptyset\}\subseteq Cx\); \(P_2(x)= m^{-0}(Cmx)\subseteq Cx\); \(P_3(x)= m^{-0}(Tmx)- \{\emptyset\}\subseteq^{-1}_* Cx\); \(P_4(x)= m^{-0}(Cmx)\subseteq^{-1}_* Cx\); \(P_5(x)= m^{-1}(Tmx)\subseteq Cx\); \(P_6(x)= m^{-1}(Cmx)\subset Cx\); \(P_7(x)= m^{-1}(Tmx)\subseteq^{-1}_* Cx\); \(P_8(x)= m^{-1}(Cmx)\subseteq^{-1}_* Cx\). We call an \(m\)-mapping of figure spaces \(j\)-continuous at a given point \(a\in X\) if \(P_j(a)\) is a true statement and \(j\)-continuous if it is \(j\)-continuous at each point \(x\in X\). Thus the author in Section 3 obtains eight types of continuities for mappings of figure spaces. These definitions immediately imply the logical relationship between the basic types with the help of a Hesse diagram. Also, the author demonstrates the logical equivalence of some basic types of continuities for \(m\)-mappings of spaces of open figures and, hence, topological spaces, which means that, in these cases, only five basic types of continuities and four derived types of continuities remain whose logical relationships is represented by a Hesse diagram. Ordered sets of \(s\)-mappings and also \(m\)-mappings of this type of continuity are investigated in Section 4. Superposition of multi-valued continuous mappings is investigated in Section 5.