NOTES ON SPACES WITH CERTAIN POINT-COUNTABLE FAMILIES

In this paper, we discuss some properties of spaces with certain point-countable families and the following examples are provided, which answer several questions posed by T. Banakh and L. Zdomskyy in [5], and S. Lin and Z.Y. Cai in [22] respectively:(1) There exists a quotient s -image of a metric space which does not have countable pseudocharacter.(2) There exists a closed and non-boundary -s -mapping f : X -> Y such that X has a compact-countable base and Y contains no closed copy of S-omega 1.(3) There is a sequence-covering mapping f from a space X with a countable sn-network onto a space Y such that f is not a 1-sequence-covering mapping.And following result is proved:Let {X-n}(n is an element of N) be a family of k-spaces with point-countable k -networks. Then every subspace with countable fan-tightness in the product space Pi(n is an element of N) Xn has a point-countable base.Finally, we give a negative answer to Cai and Lin's [7] question under property CH.