Studies On The Properties Of Base-countably Submetacompact Space And Local-strongly Submetacompact Space

This paper studies the hereditary properties,the mapping properties,and the product properties of base-countably Submetacompact space and local-Strongly Submetacompactness space. The following main results:Theorem1M is a closed subset of the base-countably submetacompact space X,meeting ω(X)=ω(M),then the space M is base-countably submetacompact space.Theorem2If the topological space X is Fσ set, and each closed subset are relative to the base-countably submetacompactness, then X is known as base-countably submetacompact space.Theorem3If the space X is base-countably submetacompact space,and the space M is the Fσ set of the space X,meeting ω(X)=ω(M),then the space M is the base-countably submetacompact space.Theorem4The following statements are equivalent:(1) The space X is base-countably submetacompact space;(2) having an open basis (?), satisfied the relation|(?)|=ω(X),(?)={(?)i}i∈N is an arbitrary countable open cover of the space X,for (?), s.t.(?)’={(?)i}i∈N is the open refinement of the (?), and (?),for (?)x∈X,(?)i∈N,s.t.,|(?)|<ω established.Theorem5If the space X is base-regularly,and X is countable submetacompact space,then the space X is base-countably submetacompact space.Theorem6The following statements are equivalent: (1)The space X is a regular base-countably submetacompact space;(2)Having an basis (?), tsatisfied the relationship|(?)|=ω(X),(?) is an arbitrary countable open cover of the space X,for (?),so that there exists a point limited shrinkage, which is composed of the elements of the (?).Theorem7The perfect mapping keeps base-countably submetacompactmess..Theorem8When the space is T2, both open and closed finite-to-one mapping f:X→Y keeps the base-countably submetacompactness.Theorem9The base-countably submetacompact mapping f:X→Y,when ω(X)≥ω(Y) and the space Y is base-countably submetacompact space,then obtained the conclusion X is base-countably submetacompact space.Theorem10The product of base-countably submetacompact space and locally-compact base-countably submetacompact space must be the base-countably submetacompact space.Theorem11If X is the i-type locally-strongly submetacompact space, then(Ⅰ)(3)(?)(2)(?)(1)；(Ⅱ)In the conditions that X is a regular space,the three locally-strongly submetacompact space are equivalent.Theorem12The closed subspace of the three locally-strongly Submetacompact space is hereditary.Theorem13If X is a regular space,for closed Lindelof mapping f:X→Y, when Y is the strongly submetacompact space, then there is also that X is strongly submetacompact space.Theorem14The open and perfect mapping hold the i-type locally-strongly submetacompactness.Theorem15In the conditions that X is a regular space, when X is i-type locally-strongly submetacompace space, Y is compact space,then can get that their product is also i-type locally-strongly submetacompact space.Theorem16The space X is i-type locally-strongly submetacompact space,the topological space Y is i-type locally compact space,then can be obtained their product is also the i-type locally-strongly submetacompact space.