Characterization Of Supra Functions On Some Supra Topological Spaces

Real-valued functions in topological spaces are important contents in general topology.Many space classes can be represented or directly defined by real-valued functions with specific conditions.In this paper,some supra topological properties of supra topological spaces are characterized by supra functions,and the definitions of supra countable paracompact spaces and supra countable compact spaces are given.This paper is divided into three chapters.In chapter 1,we mainly introduces the basic definitions and theorems of supra topological spaces.In chapter 2,we give the characterization of supra functions in supra countable paracompact spaces.In chapter 3,we give the characterization of supra functions in supra countable paracompact spaces.In this paper,the following results are obtained:Result 1 A space X is supra countably paracompact if and only if it has property(sUsL)mswl.Result 2 Let X be a supra topological space.The following are equivalent:(?)X is supra countably paracompact;(?)There is a mapping?:sL+(X)?sUSC(X)such thata)For each h ?sL+(X),0<?(h)<h,b)For each x?X,there is a supra open set V of x and m?Z such that ?(h)(y)>1/2m.(?)There is a mapping ?:sL+(X)?sUSC(X)such thata)For each<g,h>?sUsL+(X),g<?(<g,h>)<h,b)For each x?X,there is a supra open set V of x and m?Z such that?(<g,h>)(y)>1/2m for all y?V.Result 3 Let X be a supra topological space.The following are equivalent:(?)X is a supra countably compact;(?)For every decreasing sequence {F:n?Z} of supra closed subsets of X with empty intersection,there exists a decreasing sequence ?An:n?Z} of supra closed subsets of X such that Fn(?)An for each n?Z and Am=(?)for some m?Z.(?)For every decreasing sequence{Fn:n?Z} of supra closed subsets of X with empty intersection,there exists a decreasing sequence {Bn:n?Z} of supra subsets of X such that Fn(?)Bn for each n?Z and Bm=(?)for some m?Z.Result 4 X is supra countably compact(?)X has property sp(sUSC)msu(?)X has property sp(sUsL)msu(?)X has property sp(sUR)msu.