The Continuous Multifunction Spaces Weaker Than The First Countability

In this paper, we study some topological properties of continuous multifunction spaces Ck(X.R) which are weaker than the first countability. There are four chapters in the paper.In chapter one. we introduce the definitions, notations and preliminaries of continuous multifunction space Ck(X.R). We give countably strong fan tightness. Frechet property, strict Frechet property, strongly Frechet properly and k-Frechet Urysohn property on Ck(X,R).In chapter two, we discuss the equivalent conditions of countably strong fan tightness on multifunction space Ck(X,R) with compact-open topology, and by countable open k-cover sequences, we give the characteristics of countably strong fan tightness on continuous multifunction space Ck (X.R). We obtain a dual theorem of spaces X and Ck(X,R) which generalize some results of continuous single-valued function spaces to continuous multifunction spaces. We obtain the following results. Theorem 1 For a space X, the followings are equivalent:(1)sft(Ck(X))=N0;(2)sft(Ckω(X))=N0;(3) For a space X, each open k-cover sequences{(?)n}n∈N,there exists Un∈(?)n such that{Un}n∈N is an open K-cover of XIn chapter three, we give a dual theorem of Frechet property on multifunction space Ck(X.R) with compact-open topology by mean of K-sequences. We prove the equivalence of Frechet property, strict Frechet property and strong Frechet property on Ck(X,R). We generalize some results on the continuous single-valued function spaces to the continuous multifunction spaces. We obtain the following results.Theorem 2 For a space X, the followings are equivalent:(1) Ck(X) is a Frechet space;(2) Every open k-cover (?) of space X contains a k-sequence.Theorem 3 For a space X, the followings are equivalent:(1) Ck(X) is a strict Frechet space;(2) Ck(X) is a strongly Frechet space;(3) Ck(X) is a Frechet space;(4) For every open k-cover {(?)n}n∈N of space X, there exists Un∈(?)n such that{Un}n∈N is a k-sequence of X(5) Ckω(X) is a strict Frechet space.In chapter four, we discuss the dualities of k-Frechet Urysohn Property of multifunction space Ck (X, R) endowed with compact-open topology. By means of definitions of moving off and strongly compact-finite, we obtained the equivalence of Ck (X) being k-Frechet Urysohn space. We generalized some results from continuous single-valued function spaces to continuous multifunction space Ck (X.R). We obtain the following results.Theorem 4 For a space X, the followings are equivalent:(1) Ck(X) is a countable fan tightness space;(2)For every family of moving off compact sets of X, there exists a countable subfamily of strong moving off compact sets.Theorem 5 For a space X, the followings are equivalent:(1) Ck(X) is a k-Frechet Urysohn space;(2) The sequential closure of any open set Ck(X) is a closed set; (3) For every family of moving off compact sets of X, there exists a countable subfamily which is strongly compact finite;(4) For every sequence{(?)n}n∈ωof moving off compact sets of X,there exists Kn∈(?)n such that{Kn}n∈ωis strongly compact finite.