Function Characterizations Of Some Covering Properties

Real-valued functions are useful tools for the characterization of some topological spaces.Many classes of spaces can be characterized with real-valued functions that satisfy certain conditions,such as normal spaces,stratifiable spaces,semi-stratifiable spaces,etc.For example,in [24],E.Lane,P.Nyikos and C.Pan gave a characterization of stratifiable spaces: A space X is stratifiable if and only if there is an order-preserving mapψ;L(X)→C(X) such that for any h∈L(X),ψ(h)≤h and 0＜ψ(h)(x)＜h(x) whenever h(x)＞0.In this paper,we present some characterizations of countably compact spaces,pseudo-compact spaces and k-MCM spaces in terms of real-valued functions.With these results,some insertion theorems of these spaces are obtained.The paper is divided into three parts.The first chapter deals with the background of the research and the basic notions and notations.In chapter 2,some characterizations of countably compact spaces and pseudocompact spaces are given and insertion theorems of these spaces are obtained with them.The main results are:(1)For a space X,the following are equivalent:(a)X is countably compact;(b)X has property ~p(UC)_m~u;(c)X has propertyu ~p(UL)_m~u;(d)X has propertyu ~p(UR)_m~u.(2)A space X is countably compact if and only if there is a mapping ψ;L~+(X)→C(X) such that for each h∈L~+(X),ψ(h)＜h and inf{ψ(h)(x):x∈X}＞0.In chapter 3,we present a characterization of k-MCM spaces.With this result,an insertion theorem of k-MCM spaces are obtained.The main results is: X is a k-MCM space if and only if there exists a mapping φ:N×u(X)→L(X) such that(1)For each 〈f_j〉∈u(X) and n∈N,f_n≤φ(n,〈f_j〉);(2)If 〈f_j〉≤〈g_j〉,then φ(n,〈f_j〉)≤φ(n,〈g_j〉) for all n∈N,;(3)For each compact subset K of φ(n,〈f_j〉)(?)0.