Point-countable covers and sequence-covering 5-mappings at subsets

摘要

A.V. Arhangel'skii introduced the notion of an almost s-mapping. It is known that the open almost s-images of metric spaces coincide with the open boundary s-images of metric spaces. In this paper, we investigate some questions related to the sequence-covering almost s-images and sequence-covering boundary s-images of metric spaces. We establish some new characterizations of the images of metric spaces under sequence-covering and s-mappings at subsets in topological spaces.The following main results are obtained:(1) A space X has a cs*-network which is point-countable at non-isolated points if and only if X is a sequentially quotient and almost s-image of a metric space.(2) A space X has a cs-network which is point-countable at non-isolated points if and only if X is a sequence-covering and almost s-image of a metric space.(3) A space X has an sn-network which is point-countable at non-isolated points if and only if X is a 1-sequence-covering and almost s-image of a metric space.(4) A space X is a cs f -countable space if and only if X is a sequence-covering (resp., sequentially quotient) and boundary s-image of a metric space.(5) A space X is an snf-countable space if and only if X is a 1-sequence-covering and boundary s-image of a metric space. (C) 2020 Elsevier B.V. All rights reserved.