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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.

54H11 Topological groups (topological aspects)
54H13 Topological fields, rings, etc. (topological aspects)
54H20 Topological dynamics (MSC2010)
Full Text: DOI
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