We prove that a Hausdorff paratopological group G is meager if and only if there are a nowhere dense subset A ⊂ G and a countable set C ⊂ G such that CA = G = AC.2010 MSC: 22A05, 22A30.

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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.

A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. From these results, others follow on the basis of various topological assumptions. Amongst these: (i) if X is a compact Hausdorff space, then the character of the free abelian topological group on X lies between w(X) and w(X) ℵ 0 , where w(X) denotes the weight of X; (ii) if the Tychonoff space X is not a P-space, then the character of the free abelian topological group is bounded below by the "small cardinal" d; and (iii) if X is an infinite compact metrizable space, then the character is precisely d. In the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. It is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is ω-narrow. A sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces.

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