中文摘要：

本文通过集值映射扩张刻画了k-半层空间和k-MCM空间.证明了以下结果：对于空间X下列论断等价：（1）X是k-半层空间;（2）对每个度量空间Y,存在保序算子Φ使得对每个集值映射φ：X→F（Y）都对应下半连续和k-上半连续集值映射Φ（φ）：X→F（Y）,使得Φ（φ）（x）在每个点x∈Uφ有界并且φ Φ（φ）,这里F（Y）是Y的所有非空闭集,Uφ={x∈X：φ在点x局部有界}.

英文摘要：

In this paper,we give some characterizations of k-semistratifiable and k-MCM by expansions of setvalued mappings.It is shown that for a space X,the following statements are equivalent：（1）Xis k-semistratifiable;（2）for every metric space Y,there exists an order-preserving operatorΦthat assigns each set-valued mappingφ：X →F（Y）（F（Y）is the set of all nonempty closed set of Y）,a l.s.c.and k-u.s.c.set-valued mappingΦ（φ）：X →F（Y）such thatΦ（φ）（x）is bounded for each x∈Uφ,where Uφ= {x∈X：φis locally bounded at x},and thatφ Φ（φ）.