Continuous convergence of mappings (Q1901897)

The author uses Robinson-styled nonstandard analysis to study properties of the set of all multivalued functions, \({\mathcal F}\), considered as subsets of \(X\times Y\), where usually \(X\) and \(Y\) are pre-topological convergence spaces. The author's major interest is relative to the concepts of upper (UC) and lower (LC) continuity. Such a relation is said to be continuous (C) if it is both upper and lower continuous. These three concepts are defined in terms of the monad generating maps \(\rho\) and \(\sigma\) considered as functions defined on \(X\times {}^* X\), \(Y\times {}^* Y\), respectively and represented as subsets of such products. For pre-topological or topological spaces, these are the usual monads associated with such spaces. For example, \(f\in {\mathcal F}\) is upper continuous (i.e. \(f\in {\mathcal F}_{\text{UC}}\)) if \({}^* f\circ\rho \subset \sigma\circ f\). I point out that when such functions \(f\) are single-valued, then some of the author's results overlap with known results relative to the nonstandard approach to pseudo- and pre-topological convergence spaces. The major results are in terms of convergence type structures defined on \({\mathcal F}\). For example, if \((f, h)\in {\mathcal F}\times {\mathcal F}\), then \((f,h)\in \gamma_{\text{UC}}\) iff \(h\circ \rho\subset \sigma\circ f\) and, in this case, is called an upper continuous convergence on \({\mathcal F}\). Let \(\lambda: {\mathcal F}\times X\) be defined for each \(x\) in the domain of \(f\) by letting \(\lambda (f,x)= f(x)\). Then as an example of these concepts, it is established that continuous convergence \(\gamma_{\text{UC}}\) is the weakest convergence of \({\mathcal F}_{\text{UC}}\) for which \(\lambda\) is continuous in the corresponding sense. The author establishes many other interesting and useful results.