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

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

2008-01-11

arxiv.org/abs/0801.1717v2

Mathematics

General Topology

11 pages

Scientific paper

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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