Weak countability axioms of coset spaces

Abstract

In this paper, we investigate weak countability axioms of coset spaces. We first discuss biradial coset spaces. It is mainly shown that if H is a closed neutral subgroup of a topological group G, then (1) G/H is biradial ⇔ G/H is nested; (2) G/H is monotonically normal if G/H is nested; (3) G/H is metrizable ⇔ G/H is a biradial space with countable pseudocharacter. We also study coset spaces with certain point-countable covers. In particular, we prove that if H is a closed neutral subgroup of a topological group G, then (4) G/H is metrizable ⇔ G/H is determined by a point-finite cover consisting of bisequential spaces. In the end, we consider coset spaces with certain local countable networks. It is mainly shown that if H is a closed neutral subgroup of a topological group G, then (5) G/H is metrizable ⇔ G/H is an α4 and k-space with an ωω-base; (6) G/H is cosmic ⇔ G/H is a separable space with countable cn-character; (7) G/H is metrizable ⇔ G/H has countable cn-character provided G/H has the Baire property.