Some Properties Of Base-countably Weakly ? Refinable Spaces And Nearly Mesocompact Spaces

In this paper, Base-Countably weakly ? refinable spaces and Nearly-Meso compact spaces are introduced, and the closed hereditary, Tychonoff product and mapping properties of these two kinds of spaces are studied. The following main results were obtained:(i) {Fi}i?N=?n?N An is a closed cover of X , for each x? X, there existsn?N , such that 1?ord(x,An)(?)?,and each Fi(i?N) is a closed base-countably weakly ? refinable space relative to X , then X is a base-countably weakly ? refinable space .(ii) Let Y be a base -countably weakly ? refinable space and f:X?Y be a base-countably weakly ? refinable mapping and ?( X) ? ?( Y) , if Y is regular base-countably weakly ? refinable space then X is base-countably weakly ?refinable .(iii)Every closed subspace of Nearly mesocompact spaces is nearly mesocompact.(iv)A space X is nearly mesocompact if and only if every monotone open cover U has an open refinement that is compact-finite on some dense subset of X .(V )Let X = ???? X? be a |?| - paracompact space , then X is nearly mesocompactspace if and only if ???? X? is nearly mesocompact for each F?[?](?)?.(vi)A countable compact, nearly mesocompact T3-space is compact.