CHARACTERISTICS OF SPACES WITH LOCALLY COUNTABLE CS-NETWORKS VIA IMAGES OF LOCALLY SEPARABLE METRIC SPACES

Abstract

In [2], Tran Van An and Luong Quoc Tuyen proved that sn-symmetric spaces with countable cs-networks are equivalent with pseudo-sequence-covering compact images of separable metric spaces, equivalent with quotient-sequentially -images of separable metric spaces, and Cauchy sn-symmetric spaces with countable cs-networks are equivalent with 1-sequence-covering compact-covering compact images of separable metric spaces, equivalent with sequence-covering -images of a separable metric spaces. Besides, in [6, 7], Luong Quoc Tuyen proved that sn-symmetric spaces with locally countable cs-networks are equivalent with compact-covering compact ss-images of locally separable metric spaces, and Cauchy sn-symmetric spaces with locally countable cs-networks are equivalent with sequence-covering compact-covering compact ss-images of locally separable metric spaces. In this artcle, we prove that spaces with locally countable cs-networks are equivalent with sequence-covering compact-coveringss-images of locally separable metric spaces.