Paratopological and semitopological groups versus topological groups (Q555797)

A group \(G\) with a topology is called a \textit{semitopological group} if the multiplication is separately continuous, and \(G\) is called a \textit{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.