Is $\mathbb R^2\backslash \{0\}$ a manifold?

Question

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.

Answer 1

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 $\mathbb R^2$.