Paratopological and semitopological groups versus topological groups

Several new facts concerning topologies of paratopological and semitopological groups are established. It is proved that every symmetrizable paratopological group with the Baire property is a topological group. If a paratopological group G is the preimage under a perfect homomorphism of a topological group, then G is also a topological group. If a paratopological group G is a dense Gδ-subset of a regular pseudocompact space X, then G is a topological group. If a paratopological group H is an image of a totally bounded topological group G under a continuous homomorphism, then H is also a topological group. If a first countable semitopological group G is Gδ-dense in some Hausdorff compactification of G, then G is a topological group metrizable by a complete metric. We also establish certain new connections between cardinal invariants in paratopological and semitopological groups. In particular, it is proved that if G is a bisequential paratopological group such that G×G is Lindelöf, then G has a countable network. Under (CH), we prove that if G is a separable first countable paratopological group such that G×G is normal, then G has a countable base. This sheds a new light on why the square of the Sorgenfrey line is not normal.