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Continuous actions of pseudocompact groups and axioms of topological group. (English) Zbl 0786.22002
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 $$X\not\in{\mathcal N}$$.