A non-separable Christensen's theorem and set tri-quotient maps

Abstract

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 $F:\mathcal{K}(X)\to \mathcal{K}(Y)$ 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.

11 pages