Tychonoff Products Of σ-ortho Compact Space And Characterizations Of Base-Countably Paracompact Space

This paper is made of two parts:one part is the tychonoff infinite productproperties ofσ-ortho compact;In the other part,we introduce the notion ofbase-countably paracompact, and research its properties and equivalentcharacterization theorems.In this paper, we has gained the following results:1,Let X=Π_{σ∈∑}X_σbe |∑|-paracompact, X isσ-ortho compact (?)Π_σ∈FX_σisσ-ortho compact for every F∈[∑]^＜ω.2,(1) Base-countably paracompactness is an inverse invariant of quasi-perfectmappings.(2) Let X is T₂-space and Y is normal, f:X→Y be a base-paracompactmapping andω(X)≥ω(Y).If Y is base-countably paracompact, then X isbase-countably paracompact.3,(1) Let X be base-countably paracompact. If M is a closed subset of Xwithω(M)=ω(X), then M is base-countably paracompact.(2) Let X is normal countably paracompact and X=∪_i＜ωF_i, every F_i isclosed and base-countably paracompact relative to X, then X is base-countablyparacompact.4,(1) For a normal space, the following statements are equivalent:â‘ X isbase-countably paracompact;â‘¡there is a base (?) for X with |(?)|=ω(X) suchthat every countably open cover (?) of X has a locally finite refinement byclosures of member of (?).(2) For a normal space, the following statements are equivalent:â‘ X isbase-countably paracompact;â‘¡X is countably paracompact, and there is a base(?) for X with |(?)|=ω(X), such that every binary open cover of X has alocally finite refinement by members of (?);â‘¢there is a base (?) for X with|(?)|=ω(X) satisfying the following condition: for every countably open cover (?)of X, there is a locally finite cover (?)' of X by members of (?) such that(?) refines.