Nearly Base Sub-meta Compact Spaces

In this paper, we mainly study the heredity, the product and the mapping properties of nearly base sub-meta compact spaces. The following results are proved:Theorem 1 Every closed subspace of a nearly base sub-meta compact space is near base sub-meta compact.Theorem 2 Let X be a topological space and it is a F? set. If every closed subset is nearly base sub-meta compact relatively to X, then X is nearly base sub-meta compact.Theorem 3 Let X be a nearly base sub-meta compact space and M is a subset of it. If M is a F? set and ?(X)=?(M), then M is also nearly base sub-meta compact.Theorem 4 Let X be a nearly base sub-meta compact space. If every open subset of X is nearly base sub-meta compact, then every subset of X is also nearly base sub-meta compact.Theorem 5 Let X be a nearly base sub-meta compact space and the mapping f of X to Y be clopen finite to one. If X is nearly base sub-meta compact, then Y is also nearly base sub-meta compact.Theorem 6 Let X be a regular space and the continuous mapping f of X to Y, which inverse is also continuous be closed Lindelof. If Y is nearly base sub-meta compact, then X is also nearly base sub-meta compact.Theorem 7 Let X be a topological space and Y be paracompact. If the mapping f of X to Y is perfect and Y is nearly base sub-meta compact, then X is also nearly base sub-meta compact.Theorem 8 Every subset of a perpectly near base sub-meta compact space is nearly base sub-meta compact. U?(?)V(n+1)?(?)Vna for any ??A and n?NTheorem 9 Let X be a nearly base sub-meta compact space and (?) is a base of X such that |B|=?(X). Then for every locally finite closed set{U?}??A of X there exists a consequence of open sets<(?)n={Vn?:??A}>n?N and a dense set of X such that(1)U?(?)V(n+1)?(?)Vn? for any ??A and n?N;(2)For any x?D there exists a natural number n such than 1?ord(x,(?)n)<?.Theorem 10 Let X be nearly base sub-meta compact and Y be compact, then the tychonoff product X×Y is nearly base sub-meta compact.Theorem 11 Let X=????X? be a |?|-paracompact space. Then X is nearly base sub-meta compact if and only if ??F? X? is nearly base sub-meta compact for each (?)F?[?]<?Theorem 12 Let X=????X? be a countably paracompact space, then the following propositions are equivalent.(1) X is nearly base sub-meta compact;(2) ???F X? is nearly base sub-meta compact for each (?)F?[?]<?(3) ???F X? is nearly base sub-meta compact for each (?)n??.Theorem 13 Every closed nearly sub-meta compact space is developable.