Is $\mathbb R\backslash \{0\}$ a manifold ? Is $\mathbb R^2\backslash \{0\}$ a manifold ? I would say yes, but in the doubt, I prefer to ask.
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what makes you believe that the answer is "yes"? – tired Jul 07 '16 at 08:21
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evry open look homeomorphic to $\mathbb R$ (resp. $\mathbb R^2$). But I'm sure that the fact that $0$ is not in can gives some bad property. – user330587 Jul 07 '16 at 08:22
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1Every open subset of a manifold is a manifold. – user347489 Jul 07 '16 at 08:54
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Is it Hausdorff? Is it 2nd-countable? Is it locally Euclidean? – Jeff Davis Jul 07 '16 at 15:03
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You can prove that it is from the definition.
Take $x\in\mathbb R\setminus \{0\}$. So, $x$ is not $0$. Now, can you find some open set homeomorphic to $\mathbb R$ which includes $x$?
Remember, $(a,b)$ is homeomorphic to $\mathbb R$, so all you need to do is find some interval that includes $x$ but doesn't include $0$. For example, I would look for intervals centered around $x$, i.e. $(x-r, x+r)$ for some $r>0$.
For $\mathbb R^2$, remember that open balls are homeomorphic to $\mathbb R^2$.
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