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The Ellis theorem and continuity in groups. (English) Zbl 0827.54018
This paper is an interesting contribution to the existing body of theorems deducing joint continuity of an action from separate continuity. The first results in this direction go back to René Baire himself, and all subsequent ones were established under the assumption that the spaces involved are Baire spaces satisfying some sort of topological completeness condition. In the present case the author deals with the class \({\mathcal C}\) of Baire spaces which are also paracompact \(p\)-spaces. (Note that for paracompact spaces the notions of \(p\)-space, \(\omega \Delta\)-space \(M\)-space coincide; every \(p\)-space is a \(k\)-space. All Moore spaces are \(p\)-spaces.) The main result of the paper is that a separately continuous action of a left topological group \(G\) on a space \(X\) is jointly continuous if both \(G\) and \(X\) belong to the class \(\mathcal C\). The proof is based on topological game theory, following the lines of J. P. R. Christensen’s paper [Proc. Am. Math. Soc. 82, 455-461 (1981; Zbl 0472.54007)].

MSC:
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
22A20 Analysis on topological semigroups
57S25 Groups acting on specific manifolds
54H15 Transformation groups and semigroups (topological aspects)
54E52 Baire category, Baire spaces
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References:
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