Semi-continuous multifunctions and bases of countable order
A base \({\mathcal B}\) for a topological space \(X\) is called a base of countable order, if for every \(x\in X\) and every strictly decreasing sequence \(U_n\) of elements of \({\mathcal B}\) containing \(x\), the \(U_n\) create a local base at \(x\). By a famous result of Arkhangel'skij, \(X\) is metrizable if and only if it is paracompact and has a base of such kind. Let \(X\) be a regular space possessing such a base \({\mathcal B}\) and \(Y\) be a paracompact space. It is proved that an lsc closed-valued multifunction \(F:Y\multimap X\) has an usc multiselector \(H:Y\multimap X\) and lsc multiselector \(G:Y\multimap X\), both compact-valued, such that \(G\subset H\subset F\), under the condition that \({\mathcal B}\) is monotonically complete on \(F(y)\) for every \(y\in Y\) (that means: for every decreasing sequence \(U_n\) of elements of \({\mathcal B}\) such that \(U_n\cap F(y) \neq\emptyset\) for every \(n=1,2,\dots\) the intersection of all closures of \(U_n\)'s is nonempty. An application concerning compact-covering maps is also presented. The main result is a substantial generalization of an important multiselector theorem of \textit{E. Michael} [Duke Math. J. 26, 647-651 (1959; Zbl 0151.30805)].
scientific article; zbMATH DE number 1462689
scientific article