On quasi-uniform convergent sequences of multivalued maps. (English) Zbl 0759.54010

Ewert, Janina

Bull. Pol. Acad. Sci., Math. 37, No. 1-6, 137-143 (1989).

Summary: Let \(X\) be a topological space, \((Y,{\mathfrak U})\) a uniform space and \(({\mathcal Z}(Y),\tilde{\mathfrak U})\) the space of all nonempty compact subsets of \(Y\) with the induced uniformity. A sequence \(\{F_ n: n\geq 1\}\) of multivalued maps of \(X\) with values in \({\mathcal Z}(Y)\) is called quasi- uniformly convergent to a multivalued map \(F\) if for every \(U\in {\mathfrak U}\) and \(n\geq 1\) there exists a natural number \(k\) such that for each \(x\in X\) there is \(j\in\{0,1,\dots,k\}\) for which there holds \(F_{n+j}(x)\subset U[F(x)]\) and \(F(x)\subset U[F_{n+j}(x)]\). The quasi-uniform convergence preserves the upper and lower semicontinuity, and under some assumptions on \(X\) and \(Y\), also the measurability of the Baire class \(\alpha\). If \(X\) is a Baire space and \(F_ n\) are upper or lower \(\mathfrak U\)-quasi-continuous, then the function \(F: X\to({\mathcal Z}(Y),\tilde{\mathfrak U})\) is cliquish.

MSC:

54C60	Set-valued maps in general topology
54E15	Uniform structures and generalizations
54E52	Baire category, Baire spaces
54B20	Hyperspaces in general topology

Keywords:

quasi-uniform convergence