Upper Co-p-set Topologies Of Hyperspaces And Selection Principles

In this paper, we gave the definition of p-covers of the hyperspace 2X of all the closed subsets of a basic space X. By using p-cover, we obtained selection principles and covering properties of hyperspace 2X under upper co-p-set topology Tp+. There are three chapters.In chapter one, we first introduced the definitions of p-set and upper co-p-set topology Tp. We prove that a p-set is closed subset of a space X. Second, we gave the definition of p-covers on hyperspace (2X,τp-) and introduced the notations and preliminaries of a space X.In chapter two, we studied Rotherberg-like, Menger-like and Hurewicz-like selection principles of hyperspace (2X,τp-).For Rotherberg-like selection principles of hyperspace (2x,τp-), we obtained the following important results:Theorem 1 For a space X, the following are equivalent:(1) (2x,τp+) has countable strongly fan tightness;(2) Each open set Y∈X satisfies S1(Op,Op).Theorem 2 For a space X, the following are equivalent:(1) 2X satisfies S1(D≌p+2X,Dτ+之X);(2) X satisfies S1(OP,OP).Theorem 3 For a space X, the following are equivalent:(1) 2X satisfies S1(ΩAτP+,ΩAτF+) for each A∈2X;(2) Each open set Y ∈ X satisfies S1(OP,OF).For Menger-like selection principles of hyperspace (2X,τP+), we obtained the following important results:Theorem 4 For a space X, the following are equivalent:(1) (2X,τP+) has countable fan tightness;(2) Each open set Y∈X satisfies Sfin(OP, OP).Theorem 5 For a space X, the following are equivalent:(1) 2X satisfies Sfin(DτP+2X,DτP+2X);(2) X satisfies Sfin(OP,OP).For Hurewicz-like selection principles of hyperspace (2X,τP+), we obtained the following important results:Theorem 6 For a space X, the following are equivalent:(1) 2X satisfies S1(DτP+2X,DτP+gp);(2) X satisfies S1 (OP, OPgp).Theorem 7 For a space X, the following are equivalent:(1) 2X satisfies Sfin(DτP+2X,DτP+gp);(2) X satisfies Sfin (OP,OPgp).In chapter three, we discussed covering properties on (2X,τP+) and (P(X), τV+). We obtained the important results:Theorem 8 For a space X, the following are equivalent:(1) (2X,τP+) has countable set-tightness;(2) For each open set Y(?)X and each p-cover U of Y there is a countable collection {Un:n∈N} of subsets of U such that no Ui is a p-cover of Y and ∪n∈N Un is a p-cover of Y.Theorem 9 For a space X, the following are equivalent:(1) (2X,τP+) has countable T-tightness;(2) Each open set Y(?)X satisfies T(Op).Theorem 10 For a p-Lindelof space X, the following are equivalent:(1) (P(X), V+) satisfies α2(ΩP(X),ΓP(x));(2) (P(X),V+) satisfies α3(ΩP(X),ΓP(X);(3) (P(X),V+) satisfies α4(ΩP(X),ΓP(X));(4) (P(X),V+) satisfies S1(ΩP(X),ΓP(X));(5) X satisfies S1(Op,Γp)...