Abstract

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 G δ -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 G δ -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.

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