| This paper use the covering and mapping methods to preliminary research the countable mesocompact spaces,locally mesocompact spaces,countable metacompact spaces, and has gained the following results.1 (1) the followings are equivalent:(a) X is countable mesocompact spaces;(b) Let {Ui}i∈N be any countably open cover of X. Then there existscompact-finite open cover {Vi}i∈N such that Vi (?) Ui for each i∈N(2) the followings are equivalent:(a) X is countable mesocompact spaces;(b) Let {Ui}i∈N be any countably directed open cover of X. Then there existsclosure-preserving closed refinement cover F of {Ui}i∈N such that every familycomposed by compact subsets of X.(3) Quasi-perfect mapping preserves countable mesocompact spaces.(4) Let f: X→Y be perfect mapping, if Y is countable mesocompact spaces, then X is countable mesocompact spaces.(5) Let f: X→Y be mesocompact mapping, if Y is countable mesocompactspaces, then X is countable mesocompact spaces.2 (1) If condition (i) space X is i-locally mesocompact space (i=1,2, 3), then(i) 3-locally mesocompact space implies 2-locally mesocompact space and 2-locally mesocompact space implies 1-locally mesocompact space.(ii) Let X be a regular space. Then the following assertions are equivalent:(a) Space X is 3-locally mesocompact. (b) Space X is 2-locally mesocompact.(c) Space X is 1-locally mesocompact.(2) (a) Let M be a closed space. If space X is 1-locally mesocompact, so is M .(b) Let M be an open or closed space. If space X is i-locally (i=2,3)mesocompact, so is M .(3) Let X be i -locally mesocompact and Y be i -locally comapct. Then X×Y is i-locally mesocompact(i=1, 2, 3).3 (1) Let f:X→Y be a metacompact mapping. If Y is a countably metacompact space, then X is countably metacompact.(2) Let f :X→Y be a quasi-perfect metacompact mapping. If Y is acountably metacompact space, then X is countably metacompact.(3) Let X be a countably metacompact space. If Y is compact, then X×Y is countably metacompact. |
