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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.

##### MSC:
 54H11 Topological groups (topological aspects) 54H13 Topological fields, rings, etc. (topological aspects) 54H20 Topological dynamics (MSC2010)
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