Some generalized countably compact properties in topological groups

Abstract

In this paper, firstly, we study strict q-spaces, strong q-spaces and point pseudocompact spaces in topological groups in terms of preimages of metrizable spaces. A continuous mapping f:X→Y is (strongly) sequential-perfect if f is closed and (f−1(F)) f−1(y) is a sequentially compact set for each (sequentially compact set F⊆Y) y∈Y. We give some characterizations as follows:(1)A topological group G is a strict q-space if and only if it is a sequential-perfect preimage of a metrizable space.(2)A topological group G is a strong q-space if and only if it is a strongly sequential-perfect preimage of a metrizable space.(3)A topological group G is a pointwise pseudocompact space if and only if there is a continuous open mapping f from G onto a metrizable space M such that f−1(F) is an r-pseudocompact set in G for each r-pseudocompact set F in M.Secondly, we study completeness of generalized countably compact topological groups in terms of preimages of completely metrizable spaces and obtain the following results:(4)A topological group G is strictly countably sieve complete if and only if G contains a closed sequentially compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is sequential-perfect.(5)A topological group G is countably sieve-s-complete if and only if G contains a closed sequentially compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is strongly sequential-perfect.(6)A topological group G is countably sieve-p-complete if and only if G contains a closed r-pseudocompact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H satisfies that π−1(F) is r-pseudocompact in G for each r-pseudocompact set F in G/H.