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Zbl 0786.22002
Korovin, Alexander V.
Continuous actions of pseudocompact groups and axioms of topological group. (English)
[J] Commentat. Math. Univ. Carol. 33, No.2, 335-343 (1992). ISSN 0010-2628

\{This article summarizes ``an essential part'' of the author's PhD thesis.\}\par For a Tychonoff space $X$ let $C\sb p(X)$ be the set of continuous real- valued functions on $X$ in the topology inherited from $\bbfR\sp X$, and let ${\cal N}$ be the class of all spaces $X$ with this property: If $Y\subseteq C\sb p(X)$ and $Y$ is a continuous image of $C\sb p(X)$, then $Y$ has compact closure in $C\sb p(X)$. The author gives many new results extending or suggested by the now-classical theorem of {\it R. Ellis} [Duke Math. J. 27, 119-125 (1957; Zbl 0079.166)]. Here are three:\par (A) If $G\in{\cal N}$ is a group with separately continuous multiplication, then multiplication is jointly continuous; if in addition $G$ is Abelian, then $G$ is a topological group.\par (B) If $X$ is a locally compact space or $X\in{\cal N}$, 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.)\par (C) There is a space $X$ with every power $X\sp \kappa$ pseudocompact such that $X\not\in{\cal N}$.
[W.W.Comfort (Middletown)]

MSC 2000:: *22A05 Structure of general topological groups
54H11 Topological groups (topological aspects)
54C35 Function spaces (general topology)

Keywords: topological group; paratopological group; Eberlein compact; continuous multiplication

Citations: Zbl 0079.166

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