Joint continuity in semitopological monoids and semilattices. (English) Zbl 1540.22005

Osipov, Alexander V.; Kazachenko, Konstantin

Semigroup Forum 107, No. 3, 718-731 (2023).

In this article, the authors study separately continuous actions of semitopological monoids on pseudocompact spaces. They generalize the Lawson central theorem by proving that if \(\pi:S\times X\to X\) is a separately continuous quasicontinuous action of a pseudocompact right topological monoid \(S\) on a pseudocompact space \(X\), then \(\pi\) is continuous at each point of \(\{1\}\times X\). It is proved that if \(X\) is a pseudocompact space and \(Y\) is a weak \(q_D\)-space, then \((X,Y)\) is a Grothendieck pair. The authors also study subgroups of semitopological semigroups and the continuity of the multiplication in semitopological locally convex semilattices.

Reviewer: Saak S. Gabriyelyan (Beer-Sheva)

MSC:

22A15	Structure of topological semigroups
22A26	Topological semilattices, lattices and applications
54H15	Transformation groups and semigroups (topological aspects)

Keywords:

semitopological monoid; pseudocompact space; Grothendieck pair; topological group; quasicontinuous function; separately continuous function; semitopological semilattice

References:

[1]	Al’perin, M.; Osipov, AV, Generalization of the Grothendieck’s theorem, Topol. Appl., 338 (2023) · Zbl 1532.54011 · doi:10.1016/j.topol.2023.108648
[2]	Arhangel’skii, AV, Topological Function Spaces (1992), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 0758.46026 · doi:10.1007/978-94-011-2598-7
[3]	Arhangel’skii, AV; Choban, MM, Completeness type properties of semitopological groups, and the theorems of Montgomery and Ellis, Topol. Proc., 3, 33-607 (2011) · Zbl 1213.54051
[4]	Arhangel’skii, AV; Ponomarev, VI, Fundamentals of General Topology: Problems and Exercises, Mathematics and Its Applications (2001), Dordrecht: Springer, Dordrecht
[5]	Arhangel’skii, A.; Tkachenko, M., Topological Groups and Related Structures: An Introduction to Topological Algebra (2008), Paris: Atlantis Press, Paris · Zbl 1323.22001 · doi:10.2991/978-94-91216-35-0
[6]	Asanov, MO; Velichko, NV, Compact subsets in \(C_p(X)\), Commentationes Mathematicae Universitatis Carolinae, 22, 2, 255-266 (1981) · Zbl 0491.54011
[7]	Banakh, T.; Bardyla, S., Characterizing chain-compact and chain-finite topological semilattices, Semigroup Forum, 98, 234-250 (2019) · Zbl 1416.54017 · doi:10.1007/s00233-018-9921-x
[8]	Bourbaki, N., General Topology: Chapters 1-4 (1995), Berlin, Heidelberg: Springer, Berlin, Heidelberg · doi:10.1007/978-3-642-61701-0
[9]	Choban, M.; Kenderov, P.; Moors, W., Pseudo-compact semi-topological groups, Union Bulg. Math., 41, 1, 53-60 (2012)
[10]	Engelking, R., General Topology (1989), Berlin: Heldermann, Berlin · Zbl 0684.54001
[11]	Hahn, H., Reelle Funktionen: Teil 1, Punktfunktionen, Mathematik und ihre Anwendungen in Monographien und Lehrbüchern (1932), Leipzig: Akademische Verlagsgesellschaft, Leipzig · Zbl 0005.38903
[12]	Kazachenko, K.; Osipov, AV, Various types of completeness in topologized semilattices, Semigroup Forum, 104, 358-375 (2022) · Zbl 1518.22002 · doi:10.1007/s00233-022-10259-5
[13]	Kenderov, PS; Kortezov, IS; Moors, WB, Topological games and topological groups, Topol. Appl., 109, 157-165 (2001) · Zbl 0976.22003 · doi:10.1016/S0166-8641(99)00152-2
[14]	Korovin, AV, Continuous actions of pseudocompact groups and axioms of topological group, Commentationes Mathematicae Universitatis Carolinae, 33, 2, 335-343 (1992) · Zbl 0786.22002
[15]	Lawson, JD, Joint continuity in semitopological semigroups, Illinois J. Math., 18, 2, 275-285 (1974) · Zbl 0278.22002 · doi:10.1215/ijm/1256051229
[16]	Lawson, JD, Additional notes on continuity in semitopological semigroups, Semigroup Forum, 12, 265-280 (1976) · Zbl 0327.22003 · doi:10.1007/BF02195932
[17]	Moors, WB, Semitopological groups, Bouziad spaces and topological groups, Topol. Appl., 160, 15, 2038-2048 (2013) · Zbl 1285.22007 · doi:10.1016/j.topol.2013.08.008
[18]	Namioka, I., Separate continuity and joint continuity, Pacific J. Math., 51, 2, 515-531 (1974) · Zbl 0294.54010 · doi:10.2140/pjm.1974.51.515
[19]	Neubrunn, T., Quasi-continuity, Real Anal. Exch., 14, 259-306 (1988) · Zbl 0679.26003 · doi:10.2307/44151947
[20]	Reznichenko, E., Extension of mappings from the product of pseudocompact spaces, Topol. Appl., 322 (2022) · Zbl 1529.54005 · doi:10.1016/j.topol.2022.108329
[21]	Reznichenko, E., Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topol. Appl., 59, 3, 233-244 (1994) · Zbl 0835.22001 · doi:10.1016/0166-8641(94)90021-3
[22]	Reznichenko, E.; Tkachenko, M., All countable subsets of pseudocompact quasitopological Korovin groups are closed, discrete and \(C^*\)-embedded, Topol. Appl., 341 (2024) · Zbl 1531.54030 · doi:10.1016/j.topol.2023.108728