Timeline for Uniform Embedding into Euclidean Space

6 events

when toggle format	what		by	license	comment
Jul 9, 2012 at 14:56	comment	added	Sergey Melikhov		Hannes, your question is discussed at length in Isbell's book "Uniform spaces", and in some of his preceding papers. He gives some necessary conditions which make it clear that the question is not likely to have a simple answer.
Jul 7, 2012 at 18:48	comment	added	Emil Jeřábek		Note that even if you drop the assumption that $Y$ be closed, this set cannot get too wild: the other assumptions imply $Y$ is locally compact, which means it can be written as a difference of two closed sets.
Jul 6, 2012 at 14:44	comment	added	Hannes Thiel		Emil, I agree. Every closed subset of $\mathbb{R}^n$ is complete, and uniform equivalence of metric spaces preserves completeness. Thus, when asking for spaces that are uniformly equivalent to closed subsets of Euclidean space, one has to add completeness to the list of neccessary conditions. But it is maybe more natural to drop the assumption that $Y\subset\mathbb{R}^n$ be closed.
Jul 5, 2012 at 16:17	comment	added	Emil Jeřábek		Why do you require $Y$ to be closed? $X=(0,1)\subseteq\mathbb R$ meets your conditions, but AFAICS it is not uniformly equivalent to a closed subset $Y\subseteq\mathbb R^n$ ($X$ is bounded, hence so is $Y$, hence $Y$ is compact, hence $X$ is, but it actually isn’t).
Jul 5, 2012 at 16:16	answer	added	Anton Petrunin		timeline score: 2
Jul 5, 2012 at 15:33	history	asked	Hannes Thiel	CC BY-SA 3.0