Metrizable Theorems And Compact-covering And Sequence-covering Images Of Metric Spaces

In this paper, we give the equivalent characterizations of the conditions of Frink's Lemma and prove two classical results about metrizable theorems. Then we discuss the characterizations of compact-covering and 1-sequence-covering (resp. 2-sequence-covering) images of metric spaces and obtain the characterizations of a snf -countable space in which every compact subset is metrizable. Also we give a positive answer to the following question posed by professor S. Lin: How to characterize the first countable spaces in which each compact subset is metrizable? Some equivalent conditions of a space in which each compact subset is metrizable and has a countable weak base in the space are obtained by means of sn -networks. By introducing the concept of 1-scc(resp.scc) map is, it is shown that X is a 1-scc(resp.scc) image of a metric space if and only if X is a space in which each compact subset is metrizable and has a countable sn -networks. The spaces in which each compact subset has a countable outer weak base are discussed, which generalize the classic result about compact-covering and open images of metric spaces by Michael and Nagami.