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Mar 24, 2017 at 11:26 comment added Mizar For what it's worth, it is true for $n=2$.
Mar 23, 2017 at 15:14 comment added Misha If you want $L$ to be a subpolyhedron then this is already false for domains in $R^3$: Use $U$ homeomorphic to the Whitehead manifold $W$. (A regular neighborhood of such a retract $L\subset W$ would have to be a PL 3-ball but $W$ cannot be exhausted by 3-balls.) I am not sure about general compact subsets $L$.
Mar 23, 2017 at 11:19 comment added Mark Grant Have you looked at the section of Dold's "Lectures on Algebraic Topology" on ENRs? It might give you some ideas of how to prove (or disprove) what you want.
Mar 23, 2017 at 7:28 history edited Myshkin CC BY-SA 3.0
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Mar 23, 2017 at 3:13 comment added Yemon Choi Are you making any smoothness assumptions about the boundary of U?
Mar 23, 2017 at 2:51 review First posts
Mar 23, 2017 at 7:28
Mar 23, 2017 at 2:50 history asked E.G. CC BY-SA 3.0