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Is it true, that having an open ball $B(0,1)$ in $\mathbb{R}^n$ (say $n=2$, but the more general answer the better) and a set $C$ such that $0 \in \overline{C}$, there is no open neighbourhood of $0$ in $B(0,1)\setminus C$ homeomorphic with $B(0,1)$.

This comes from something with less technical wording: is it true, that image of every embedding of an $n$-manifold into another $n$-manifold is open?

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Your question (in the title) is the content of the invariance of domain theorem of Brouwer. For a proof (using the Brouwer fixed point theorem) you can see Terry Tao's blog.

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Yes! That is what I needed. –  savick01 Jan 6 '12 at 15:16

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