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Set-valued mappings and an extension theorem for continuous functions. (English) Zbl 0611.54009
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.]
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 $$G_{\delta}$$ set containing A. A Moore space is a regular space Y which has a sequence $$\{$$ $${\mathcal C}_ n\}^{\infty}_{n=0}$$ of open covers of Y such that, for every $$y\in Y$$, $$\{$$ $$U\{$$ C:$$\in {\mathcal C}_ n$$ and $$y\in C\}\}^{\infty}_{n=0}$$ 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 $$G_{\delta}$$ 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 $$G_{\delta}$$ 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.
Reviewer: P.Morales

##### MSC:
 54C20 Extension of maps 54C60 Set-valued maps in general topology 54E30 Moore spaces