Local minimalities in paratopological groups

摘要

In this paper, we mainly discuss the cardinal invariants on some class of paratopological groups. For each i ∈ {0,1,2,3,3.5}, we define the class of locally Ti-minimal paratopological groups by the conditions that, for a T_i paratopological group (G,τ), there exists a τ-neighborhood U of the neutral element such that U fails to be a neighborhood of the neutral element in any T_i-semigroup topology on G which is strictly coarser than τ. We mainly prove that (1) each UFSS and T_i-paratopological Abelian group (G, τ) is locally T_i-minimal; (2) if (G, τ) is a regular locally T_1-minimal Abelian paratopological group then x(G) = πx(G); (3) if (G,τ) is an Abelian locally T3-minimal paratopological group then we have w(G) = nw(G). Moreover, we also discuss some relations of locally Ti-minimal paratopological groups and some properties of subgroups of T_i-minimal paratopological groups. Some questions are posed.