# Are open sets in R^n always homeomorphic to R^n?

by iLoveTopology
Tags: homeomorphic, sets
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 P: 11 I know that open intervals in R are homeomorphic to R. But does this extend to any dimension of Euclidean space? (Like an open 4-ball is it homeomorphic to R^4?) My book doesn't talk about anything general like that and only gives examples from R^2.
 Sci Advisor P: 1,172 No for general open sets; look at, e.g., an open annulus, or any disconnected open set. But an open n-disk D:={x in R^n : ||x||<1 } (or any translation of it) is homeomorphic to R^n.
 Sci Advisor P: 1,810 Any connected open set in R^n is homeomorphic to R^n, for any n. An open set in R^n is homeomorphic to the disjoint union of equally many R^n's as connected parts of your open set.
 Sci Advisor P: 1,172 Are open sets in R^n always homeomorphic to R^n? Actually, an open annulus is open and connected, but not homeomorphic to R^n, since it is not simply-connected.
Sci Advisor
P: 1,810
 Quote by Bacle2 Actually, an open annulus is open and connected, but not homeomorphic to R^n, since it is not simply-connected.
thanks for the correction, i meant simply connected :)
Mentor
P: 18,346
 Quote by disregardthat thanks for the correction, i meant simply connected :)
Also not true, then open set $B(0,1)\setminus\{0\}$ of $\mathbb{R}^3$ is simply connected but not homeomorphic to $\mathbb{R}^3$.
 Sci Advisor HW Helper P: 9,495 notice R^n is contractible. But even that is not enough. Look at this: http://math.stackexchange.com/questi...c-to-mathbb-rn

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