A note on spaces with certain point-countable networks

Abstract

We obtain two countable properties of spaces with a pointcountable sn-network, establish the mapping relation between spaces with a pointcountable wcs * -network of certain property and locally separable metric spaces, and partially correct a gap in [8].

Further results on images of metric spaces

“…A space X is an ℵ 0 -space if X is regular and it has a countable cs-network [23]. Cai and Li [5] established the mapping relation between a space with a pointcountable wcs * -network and a locally separable metric space. The main result is as follows.…”

“…Then, we systematically get characterizations of weakly continuous strong wc-images (wcimages, wk-images) of locally separable metric spaces by means of certain networks, which implies an answer to Question (1). Also, we give counterexamples to sharpen some results on images of locally separable metric spaces in [5,20], which give a negative answer for Question (2) and some more.…”

“…There are many modifications of k-networks such as cs-networks, cs * -networks, wcs * -networks, sn-networks, so-networks, etc. These notions are used to characterize continuous images of metric spaces with certain covering-properties [2, 5,7,10,19,21,[26][27][28][29]. In 2002, Lin [19] posed the following problem.…”

“…The difficulty in solving this problem is that each member P of a wcs * -network P may not contain the limit point of certain sequence which is frequently in P . Around this problem, Cai and Li [5] established a relation between spaces with point-countable wcs * -networks of certain properties and images of locally separable metric spaces; Lin and Li [20] introduced the concepts of wks-mappings and wcs-mappings, and them to characterize spaces with point-countable k-networks. Their main result is as follows.…”