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Continuous actions of pseudocompact groups and axioms of topological group. (English) Zbl 0786.22002
{This article summarizes “an essential part” of the author’s PhD thesis.}
For a Tychonoff space \(X\) let \(C_ p(X)\) be the set of continuous real- valued functions on \(X\) in the topology inherited from \(\mathbb{R}^ X\), and let \({\mathcal N}\) be the class of all spaces \(X\) with this property: If \(Y\subseteq C_ p(X)\) and \(Y\) is a continuous image of \(C_ p(X)\), then \(Y\) has compact closure in \(C_ p(X)\). The author gives many new results extending or suggested by the now-classical theorem of R. Ellis [Duke Math. J. 27, 119-125 (1957; Zbl 0079.166)]. Here are three:
(A) If \(G\in{\mathcal N}\) is a group with separately continuous multiplication, then multiplication is jointly continuous; if in addition \(G\) is Abelian, then \(G\) is a topological group.
(B) If \(X\) is a locally compact space or \(X\in{\mathcal N}\), then some “multiplication” from \(X\times X\) to \(X\) makes \(X\) a topological group if and only if some Abelian group acts transitively on \(X\). (In particular, an Abelian group cannot act transitively on a compact, non- dyadic space.)
(C) There is a space \(X\) with every power \(X^ \kappa\) pseudocompact such that \(X\not\in{\mathcal N}\).

MSC:
22A05 Structure of general topological groups
54H11 Topological groups (topological aspects)
54C35 Function spaces in general topology
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