A Product Properties Of Completely Regular Strict Quasi-Paracompact Spaces

We introduce completely regular strict quasi-paracompact spaces and their basic properties in this paper, and then mainly study their mapping properties and product, etc. The main conclusions are the followings:Theorem1Let X be a completely regular space, then X is a strict quasi-paracompact space if and only if X is quasi-paracompact.Theorem2Let X be a completely regular strict quasi-paracompact space. If F is a closed subset of X,then F is completely regular strictquasi-paracompact.Theorem3Let D be a open subspace of X.If D is a completely regular strict quasi-paracompact space, then every subspace of X is completely regular strict quasi-paracompact.Theorem4Let Y be a completely regular strict quasi-paracompact space. If mapping f:X�'Y is a closed mapping of finite to one,then X is completely regular strict quasi-paracompact.Theorem5Let f:X�'Y be a perfect mapping.If Y is a completely regular strict quasi-paracompact space, then X is completely regular strict quasi-paracompact.Theorem6Suppose X is a|Λ|-paracompact,and X is the limit of the inverse system {Xσ,πρσ,Λ} for any σ,ρ∈Λ,and we say X=Lim{Xσ,πρσ,Λ} and every projection mapping πσ:X�'Xσ is an open surjection.If any Xa is a completely regular strict quasi-paracompact space, then X is completely regular strict quasi-paracompact.Theorem7Soppose X is a countably paracompact space,and X is the limit of the inverse system {Xσ,πρσ,Λ},for any index set A and any σ,ρ∈A,and we say X=Lim{Xσ,πρσ,Λ} and every projection mapping πσ:X�'Xσ is an open surjection.If every Xσ is a completely regular strict quasi-paracompact space,then X is completely regular strict quasi-paracompact.Theorem8Let X is a completely regular strict quasi-paracompact space. If Y is compact,then X×Y is completely regular strict quasi-paracompact. Theorem9Suppose {Xα}α∈Λ is a family of some subset of a space X,and∏α∈ρXα is completely regular strict quasi-paracompact for every ρ∈Λ.If Y is a compact space,then∏α∈ΛXα×Y is completely regular strict quasi-paracompact.Theorem10Soppose {Xσ}α∈Λ is a family of some subset of a space X,and X=∏α∈ΛXσ is a|Λ|-paracompact.If∏σ∈ρXσ. is a completely regular strict quasi-paracompact for every ρ∈∑and∑=[Λ]<ω,then X is completely regular strict quasi-paracompact.Theorem11Soppose {Xσ}σ∈Λ is a family of some subset of a space X,and X=∏σ∈ΛXσ is countably paracompact space.If∏σ∈ρXσ is a completely regular strict quasi-paracompact space for every ρ∈∑and∑=[Λ]<ω,then X is completely regular strict quasi-paracompact.