×

zbMATH — the first resource for mathematics

Paratopological and semitopological groups versus topological groups. (English) Zbl 1077.54023
A group \(G\) with a topology is called a semitopological group if the multiplication is separately continuous, and \(G\) is called a paratopological group if the multiplication is jointly continuous. Clearly, every topological group is paratopological group and semitopological group. On the other hand, the Sorgenfrey line is an example of a paratopological group which is not a topological group.
In the first section of this paper, the authors prove that a paratopological group \(G\) is a topological group if \(G\) satisfies one of the following properties: (1) \(G\) is symmetrizable Hausdorff with the Baire property, (2) \(G\) is a preimage under a perfect homomorphism of a topological group, (3) \(G\) is an image of totally bounded topological group under a continuous homomorphism. They also prove that if a first countable semitopological group \(G\) is \(G_{\delta}\)-dense in some Hausdorff compactification of \(G\), then \(G\) is a topological group metrizable by a complete metric.
In the second section, they establish new connections between cardinal invariants in paratopological groups.

MSC:
54H11 Topological groups (topological aspects)
54H13 Topological fields, rings, etc. (topological aspects)
54H20 Topological dynamics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arhangel’skii, A.V., Mappings and spaces, Russian math. surveys, 21, 115-162, (1966) · Zbl 0171.43603
[2] Arhangel’skii, A.V., Bisequential spaces, tightness of products, and metrizability conditions in topological groups, Trans. Moscow math. soc., 55, 207-219, (1994) · Zbl 0842.54004
[3] Arhangel’skii, A.V.; Just, W.; Reznichenko, E.A.; Szeptycki, P., Sharp bases and weakly uniform bases versus point-countable bases, Topology appl., 100, 39-46, (2000) · Zbl 0937.54015
[4] Bouziad, A., Every čech-analytic Baire semitopological group is a topological group, Proc. amer. math. soc., 24, 3, 953-959, (1998) · Zbl 0857.22001
[5] Burke, D.K., Covering properties, (), 347-422
[6] Chen, Y.Q., Note on two questions of arhangel’skii, Questions answers gen. topology, 17, 91-94, (1999)
[7] Choban, M.M., Topological structure of subsets of topological groups and their quotients, (), 117-163, (in Russian)
[8] Ellis, R., A note on the continuity of the inverse, Proc. amer. math. soc., 8, 372-373, (1957) · Zbl 0079.04104
[9] Engelking, R., General topology, (1977), PWN Warszaw
[10] Garcia-Ferreira, S.; Garcia-Maynez, A., On weakly pseudocompact spaces, Houston J. math., 20, 1, (1994) · Zbl 0809.54012
[11] Montgomery, D., Continuity in topological groups, Bull. amer. math. soc., 42, 879-882, (1936) · JFM 62.1230.04
[12] Nagami, K., σ-spaces, Fund. math., 61, 169-192, (1969) · Zbl 0181.50701
[13] Nyikos, P.J., Metrizability and the fréchet – urysohn property in topological groups, Proc. amer. math. soc., 83, 4, (1981) · Zbl 0474.22001
[14] Reznichenko, E.A., Extensions of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology appl., 59, 233-244, (1994) · Zbl 0835.22001
[15] O. Ravskij, Private communication, 2001
[16] Roelke, W.; Dierolf, S., Uniform structures on topological groups and their quotients, (1981), McGraw-Hill New York
[17] Uspenskiıˇ, V.V., Compact factor spaces of topological groups and haydon spectra, Math. notes, 42, 827-831, (1987) · Zbl 0653.22002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.