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Applications of set-valued mappings to various spaces of continuous functions

✍ Scribed by K. Kuratowski

Publisher
Elsevier Science
Year
1971
Weight
727 KB
Volume
1
Category
Article
ISSN
0016-660X

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✦ Synopsis

Given two spaces X and Y, three kinds of continuous nppinge f : X --, Y are considered: the set * of all monotone mappings, the set .&?of idl light mappings and the set % of all non-alternating (onto) mappings. It is shown that -kmde.\P suitable .lssumptions on the spaces X and Y -the sets%, .@and % are G, sets ia the space Y*. It follows that, if Y is complete metric, then%, 2 and % are topologically complete spaces.

The most important properties of set-valued m,appings which q.re applied in proving the main theoxems of section II are collected in section I. 1 AMS Subj. Class.: 5428 1 uc tion 1. Semi-continuous sewdued mappings. Let X and Y be two topological spaces, and let F(y) be, for each y E Y, a closed subset of X. In e F : Y+ 2x, where 2" denotes the space of all endowed with the Vietoris topology; this means that &he totality of sets either of the form (K : KC G) or of the form n 4; # 0 j is a subspace of 2x (here K is a closed subset and G an

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A closed nonvoid subset Z of a Banach space X is called antiproximinal if no point outside Z has a nearest point in Z. The aim of the present paper is to prove that, for a compact Hausdorff space T and a real Banach space E, the Banach space C T E , of all continuous functions defined on T and with