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

TL;DR

This paper extends Christensen's theorem to more general spaces by exploring additional properties of set tri-quotient maps, beyond the classical separable metrizable case, to ensure completeness transfer.

Contribution

It introduces new conditions on set tri-quotient maps that allow Christensen's completeness result to hold in non-separable spaces.

Findings

Identifies properties of set tri-quotient maps that preserve completeness in broader spaces

Extends Christensen's theorem beyond separable metrizable spaces

Provides conditions under which completeness is transferred via monotone 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 $F\colon\mathcal{K}(X)\to\mathcal{K}(Y)$ 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.