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I know that open balls are homeomorphic to the entire Euclidean space, and any convex open set can be proved to be homeomorphic to the entire Euclidean space. So I was wondering if all the open sets in $\mathbb{R}^n$ are homeomorphic?

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    This is not the only obstruction. See https://math.stackexchange.com/questions/427787/characterization-of-the-subsets-of-euclidean-space-which-are-homeomorphic-to-the?rq=1 for the exact obstructions for open subsets to be homeomorphic to the space itself.2018-05-03

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No. For example, the open annulus $\{(x, y) \in \Bbb R^2 : 1 < x^2 + y^2 < 2\}$ is not simply connected, but the open ball $\{(x, y) \in \Bbb R^2 : x^2 + y^2 < 1\}$ is.