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Zbl 0611.54009
Levi, S.
Set-valued mappings and an extension theorem for continuous functions.
(English)
[A] Topology theory and applications, 5th Colloq., Eger/Hung. 1983, Colloq. Math. Soc. János Bolyai 41, 381-392 (1985).

[For the entire collection see Zbl 0588.00022.] \par Let us introduce some terminology. Let Y be a Hausdorff completely regular space. We say that Y is an extension complete space and we write $Y\in ECS$ if, for every topological space X, every dense subset A of X and every continuous function $f: A\to Y$, f has a continuous extension to a $G\sb{\delta}$ set containing A. A Moore space is a regular space Y which has a sequence $\{$ ${\cal C}\sb n\}\sp{\infty}\sb{n=0}$ of open covers of Y such that, for every $y\in Y$, $\{$ $U\{$ C:$\in {\cal C}\sb n$ and $y\in C\}\}\sp{\infty}\sb{n=0}$ is a local base at y. Then the main results can be stated as follows: Theorem 1. Let Y be a Hausdorff regular space such that its diagonal is a $G\sb{\delta}$ set in $Y\times Y$, let X be a topological space, let A be a dense subset of X and let $f: A\to Y$ be a continuous function. If for every $x\in X\setminus A$ there exists an open neighborhood U of x such that $\overline{f(A\cap U)}$ is compact, then f has a continuous extension to a residual $G\sb{\delta}$ set containing A. Theorem 2. If Y is a Čech-complete Moore space, then $Y\in ECS$. Theorem 3. The class ECS is strictly contained in the class of Čech-complete spaces and it is closed under countable products, countable intersections, closed subsets and cozero subsets.
[P.Morales]
MSC 2000:
*54C20 Extension of maps on topological spaces
54C60 Set-valued maps
54E30 Moore spaces

Keywords: extension complete space; Moore space; continuous extension; Čech- complete Moore space; countable products; countable intersections; closed subsets; cozero subsets

Citations: Zbl 0588.00022

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