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.

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