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Convex Structures and Continuous Selections

Published online by Cambridge University Press:  20 November 2018

Ernest Michael
Ernest Michael*
Affiliation:
University of Washington
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This paper continues the study of continuous selections begun in (13; 14; 15) and the expository paper (12). The purpose of these papers, which is described in detail in the introduction to (13), can be summarized here as follows. If X and Y are topological spaces, and ϕ a function (called a carrier) from X to the space 2Y of non-empty subsets of F, then a selection for ϕ is a continuous f: X → Y such that f(x) ∈ ϕ(x) for every x ∈ X. For reasons which are explained in (13), we restrict our attention to carriers which are lower semi-continuous (l.s.c), in the sense that, whenever U is open in Y, then is open in X. Our purpose in these papers is to find conditions for the existence and extendability of selections.

The principal purpose of this paper is to generalize the following result, which is half of the principal theorem (Theorem 3.2˝) of (13) (and is repeated as Theorem I of (12)).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 556 - 575
Copyright
Copyright © Canadian Mathematical Society 1959

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