The Finite Subsequence-covering Mappings And π Images

In this paper, we discuss the properties of the finite subsequence-covering mappings and relation between it and other mappings. We also obtain some inter characterizations of the π images of locally separable metric spaces.In chapter 2, we prove that sn—first countable spaces are preserved by the finite subsequence-covering mappings.By this result, we prove that the finite subsequence-covering, quotient mappings preserve g—metrizable spaces, also prove that the finite subsequence-covering, closed mappings preserve sn—metrizable spaces, g-metrizable spaces, metrizable spaces, point-countable bases. In addition, conclusions are obtained that mappings on first countable spaces are almost open mappings if and only in the mappings are one-sequence-covering, pseudo-open mappings ,and one-sequence-quotient mappings on sn—first countable spaces are one-sequence-covering mappings. At last, an example are given out to show that not all one-sequence-quotient mappings are sequence-covering mappings.In chapter 3,using the notion of sieve to give characterizations of the sequence-quotient and sequence-covering mappings π images of metric spaces and locally separable metric spaces. We prove that a space X is a sequence-quotient (sequence-covering) π images of metric spaces if and only if X has a point-star network which has cs*(cs) sieve and a space X is a sequence-quotient (sequence-covering) π images of locally separable metric spaces if and only if X has a point-star network of acountable threat which has cs*(cs) sieve.