Star Covering Space And Their Relative Properties

A space X is called star countably compact,if for every open cover ? of X,there exists a countably compact subset P of X such that St(P,?)= X,where St(P,?)= U{U??:U?P??}.A space Y(?)X is called 1(2)-star countably compact,if for every open cover ? of X,there exists a countably compact subset P of X(Y)such that Y(?)St(P,?).In this paper,we mainly discuss topological properties of star countably compact,relative star countably compact space and relative absolutely star countably compact,which include the product properties of it with compact spaces and topological properties under continuous mapping.This article is divided into four chapters.In the first chapter we mainly introduce some basic concepts and basic properties of topology space.In the second chapter we introduce the concept of star countably compact space and the product properties with the compact space.In the third chapter we introduce the concept and basic properties of relative absolute star countably compact space.In the fourth chapter we study some topological properties of relative star countably compact space.The following results are proved:(1)Let f:X?Y be a continuous map from a star countably compact space X onto a space Y.If X be a star countably compact space,then Y is a star countably compact space.(2)Let X be a star countably compact space.If F is a closed subset in X,then F is star countably compact space.(3)Let X be a star countably compact space.If Y is a compact space,then X × Y is a star countably compact space.(4)Let f be a open perfect map from a space X to a star countably compact space Y.Then X is star countably compact space.(5)If there exist a continuous one to one open mapping f:X?Y from a space X onto a space Y,and B is(strongly)absolutely star countably compact space in X,then f(B)is(strongly)absolutely star countably compact space in Y.(6)Let f:X?Y be a continuous map from space X onto a space Y.If F is 1(2)-star countably compact space in X,then f(B)is 1(2)-star countably compact space inY.(7)Let X be a topological space.If Y is 1-star countably compact space in X,Z is a compact space,then Y×Z is 1-star countably compact space in XxZ.