Some Surveys On Base-mesocompact

In this paper base-mesocompact spaces and strongly base-mesocompact spaces are investigated by mapping and covering methods and the following results are mainly obtained.1.Results on base-mesocompact spaces(1)Let X be base-mesocompact and M a closed subset of X withω(X)=ω(M),then M is base-mesocompact.(2)Let X,Y be mesocompact spaces withω(X)≥ω(Y).If X×Y is base-mesocompact,then so is X.(3)Let X be base-mesocompact and X′be an F_σsubset of X.If X is normal,then X′is base-mesocompact relative to X.(4)If X is a normal mesocompact space and the union of closed base-mesocompact spaces relative to X,then X is base-mesocompact.(5)Let X be mesocompact,locally base-mesocompact.If X is normal then X is base-mesocompact.(6)Let Y be a base-mesocompact space and f:X→Y be a base-mesocompact map andω(X)≥ω(Y).If Y is regular then X is base-mesocompact.(7)Let f:X→Y be a base-paracompact map andω(X)≥ω(Y).X is base-mesocompact if Y is regular and base-mesocompact.(8)Let f:X→Y be a closed Lindel(?)f map with regular domain and regular range andω(X)be a regular cardinality.If Y is base-mesocompact,then X is base-mesocompact.Moreover the regularity of X can not be replaced by X is Hausdorff space.(9)Let X and Y be both base-mesocompact.X×Y is base-mesocompact if Y is locally compact.2.Results on strongly base-mesocompact spaces(1)Let X be strongly base-mesocompact and M a closed subset of X withω(X)=ω(M),then M is strongly base-mesocompact.(2)Let f:X→Y be a perfect map.X is strongly base-mesocompact if Y is strongly base-mesocompact.(3)Let Y be a strongly base-mesocompact space and f:X→Y be a strongly base-mesocompact map andω(X)≥ω(Y).If Y is regular then X is strongly base-mesocompact.(4)Let X be strongly base-mesocompact and Y be a compact space.Then X×Y is strongly base-mesocompact.(5)Let X be locally compact then X is base-mesocompact if and only if X is strongly base-mesocompact.(6)Let X be a regular k-space.Then X is strongly base-mesocompact if and only if X is base-paracompact.(7)Let X be r-space.Then X is base-mesocompact if and only if X is base-paracompact.