On (para)topological groups with a countably (s-)complete sieve

Abstract

Countably sieve-complete spaces were defined by M. Michael in 1972. In this article, we introduce a notion called countably sieve-s-complete spaces. Some properties of countably sieve-complete (countably sieve-s-complete, strongly countably complete) spaces are discussed. We get the following main results.A topological group is countably sieve-complete if and only if G contains a closed countably compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is closed. By the above conclusion and a known conclusion, we get that a topological group G is strongly countably complete if and only if G is countably sieve-complete. A topological group G is countably sieve-s-complete if and only if G contains a sequentially compact closed subgroup H with a countable base of open neighborhoods such that the quotient space G/H is a completely metrizable space and the canonical quotient mapping π:G→G/H is closed. An ω-balanced topological group G is countably sieve-complete if and only if G contains a closed countably compact invariant subgroup H such that the quotient space G/H is a completely metrizable topological group and the canonical quotient mapping π:G→G/H is closed. A topological group G is ω-narrow and countably sieve-s-complete if and only if G contains a sequentially compact closed invariant subgroup H with a countable base of open neighborhoods such that the quotient space G/H is a completely metrizable second-countable topological group and the canonical quotient mapping π:G→G/H is closed.We show that if G is a regular countably sieve-complete semitopological group with Sm(G)≤ω and satisfies property (⁎), then G is a topological group. Every regular totally ω-narrow countably sieve-complete paratopological group is a topological group.