Cardinal invariants of coset spaces

Abstract

A topological space X is called a coset space if X is homeomorphic to a quotient space G/H of left cosets, for some closed subgroup H of a topological group G. In this paper, we investigate the cardinal invariants of coset spaces. We first show that if H is a closed neutral subgroup of a topological group G, then △(G/H)=ψ(G/H), w(G/H)=d(G/H)⋅χ(G/H) and w(G/H)=l(G/H)⋅χ(G/H).We also prove that if H is a closed subgroup of a feathered topological group G, then (1) w(G/H)=d(G/H)⋅χ(G/H) and w(G/H)=l(G/H)⋅χ(G/H); (2) the quotient space G/H is metrizable if and only if G/H is first-countable.At the end, we consider some applications of sp-networks in coset spaces. In particular, we show that if H is a closed neutral subgroup of a topological group G, then (1) spnw(G/H)=d(G/H)⋅spχ(G/H); (2) the quotient space G/H is metrizable if and only if G/H has countable sp-character.