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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)].

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
Full Text: DOI
[1] Arhangel’skiǐ, A.V., On a class containing all metric and all locally bicompact spaces, Dokl. akad. nauk SSSR, 151, 751-754, (1963), (in Russian)
[2] Bourbaki, N., Topologie Générale, (1961), Hermann Paris, Ch. 1 and 2 · Zbl 0139.15904
[3] Bouziad, A., Jeux topologiques et points de continuité d’une application séparément continue, C. R. acad. sci. Paris, 310, 359-361, (1990) · Zbl 0701.90110
[4] Brand, N., Another note on the continuity of the inverse, Arch. math., 39, 241-245, (1982) · Zbl 0476.22001
[5] Brown, L.G., Topological complete groups, Proc. amer. math. soc., 35, 593-600, (1972) · Zbl 0251.22001
[6] Christensen, J.P.R., Joint continuity of separately continuous functions, Proc. amer. math. soc., 82, 455-461, (1981) · Zbl 0472.54007
[7] Ellis, R., Locally compact transformation groups, Duke math. J., 24, 119-125, (1957) · Zbl 0079.16602
[8] Ellis, R., A note on continuity of the inverse, Proc. amer. math. soc., 8, 372-373, (1957) · Zbl 0079.04104
[9] Engelking, R., General topology, (1977), PWN Warsaw
[10] Gruenhage, G., Generalized metric spaces, (), 423-819
[11] Hansel, G.; Troallic, J.P., Points de continuité à gauche d’une action de semigroupe, Semigroup forum, 26, 205-214, (1983) · Zbl 0504.22003
[12] Helmer, D., Continuity of semigroup actions, Semigroup forum, 23, 153-188, (1981) · Zbl 0473.22002
[13] Lawson, J.D., Points of continuity for semigroup actions, Trans. amer. math. soc., 284, 515-531, (1984)
[14] Morita, K., A survey of the theory of M-spaces, Gen. topology appl., 1, 49-55, (1971) · Zbl 0213.24002
[15] Namioka, I., Separate continuity and joint continuity, Pacific J. math., 51, 515-531, (1974) · Zbl 0294.54010
[16] Pfister, H., Continuity of the inverse, Proc. amer. math. soc., 2, 312-314, (1985) · Zbl 0579.22001
[17] Ruppert, W., Compact semitopological semigroups: an intrinsic theory, () · Zbl 0606.22001
[18] Saint Raymond, J., Jeux topologiques et espaces de namioka, Proc. amer. math. soc., 3, 449-504, (1983) · Zbl 0511.54007
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