不可分的Christensen定理与集三商映射
A non-separable Christensen's theorem and set tri-quotient maps
摘要：
对于每个空间$X$，令$\mathcal K(X)$是$X$的所有紧子集的集合。Christensen \cite{c:74}证明了：如果$X, Y$是可分度量化空间，且$F\colon\mathcal{K}(X)\to\mathcal{K}(Y)$是单调映射，使得对于某些$K\in\mathcal{K}(X)$，任何$L\in\mathcal{K}(Y)$都被$F(K)$覆盖，则在$X$完备的条件下，$Y$是完备的.众所周知，\cite{bgp}的这个结果对不可分空间是不成立的.本文讨论了$F$的一些附加性质，这些性质保证了Christensen结果在更一般空间中的有效性。
英文摘要：
For every space $X$ let $\mathcal K(X)$ be the set of all compact subsets of $X$. Christensen \cite{c:74} proved that if $X, Y$ are separable metrizable spaces and $F\colon\mathcal{K}(X)\to\mathcal{K}(Y)$ is a monotone map such that any $L\in\mathcal{K}(Y)$ is covered by $F(K)$ for some $K\in\mathcal{K}(X)$, then $Y$ is complete provided $X$ is complete. It is well known \cite{bgp} that this result is not true for non-separable spaces. In this paper we discuss some additional properties of $F$ which guarantee the validity of Christensen's result for more general spaces.
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