Reflecting some properties of topological groups

Abstract

We show that for a number of cardinal functions ϕ, an ω-narrow topological group G admits a continuous homomorphism onto a topological group H satisfying ϕ(H)=w(H)=τ, for every infinite regular cardinal τ≤ϕ(G). In particular, the conclusion is valid when ϕ is weight, cellularity, character, pseudocharacter, tightness or o-tightness (the regularity of τ is not necessary for the cellularity). We also show that both pseudocompactness and precompactness reflect in continuous homomorphic images of countable weight, so an ω-narrow topological group is pseudocompact (precompact) if and only if all continuous second countable homomorphic images of the group are pseudocompact (precompact). It is established that the Baire property of ω-narrow groups reflects in continuous homomorphic images of weight less than or equal to the continuum.It turns out, however, that neither compactness, countable compactness, nor (weak) Lindelöfness reflects in continuous homomorphic images of any ‘small’ weight.