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Schauder's lemma asserts that you can always extend a uniformly continuous, uniformly open map from a dense subset of a complete metric space to a uniformly open map on the completion. I think the ...

Let $X$ and $Y$ be metric spaces and let $f:X\rightarrow Y$ be such that: for every compact subset $K$ of $X$ the restricted map $f|_K:K\rightarrow Y$ defined by $f|_K(x)=f(x)$ is bi-Lipschitz (with ...