Any neighbourhood of a point $x$ in a manifold $X$ ($\dim X \geq 2$) has a subneighbourhood $V$ of $x$ such that $V \setminus \{x\}$ is connected

Question

What I want to show for this is that $V \setminus \{x\}$ is homeomorphic to some punctured ball, $B_m \setminus \{p\}$ (where $B_m$, $p \in \Bbb R^m$).

And then since $B_m \setminus \{p\}$ is path-connected, $V \setminus \{x\}$ is path-connected hence connected.

So far all I've got is this: I know that since $X$ is an $m$-manifold there a homeomorphism $$\phi_\alpha : U_x \rightarrow A$$ where $U_x$ is a open neighborhood of x, $U_x \subseteq X$, and $A$ is open subset of $\Bbb R^m$.

Then take $C = \phi_\alpha(U_x)\cap A$. Then since $\phi_\alpha(U_x)$ is open and $A$ is open, $C$ is open. So then I can take a ball $B_m \subseteq C$. But this is where I get stuck.

Answer 1

You're very nearly there:

Since $A = \phi(U_x)$ is open, there is some ball $B \subseteq A$ containing $p := \phi(x)$ (because open balls form a basis of the standard topology on $\mathbb{R}^n$). If we denote $V := \phi^{-1}(B)$, the restriction $\phi\vert_V$ is a homeomorphism between $V \ni x$ and the ball $B$, and restricting further gives a homeomorphism between $\phi^{-1}(B - \{p\}) = V - \{x\}$ and the punctured ball $B - \{p\}$.

Answer 2

It can be shown that a manifold $M$ has a basis of coordinate balls since in general any neighborhood of $M$ homeomorphic to $\mathbb{R}^m$ is also homeomorphic to a ball $B_r(x)$ where $r>0$ is the radius and $x$ is the center point. This is true because $\mathbb{R}^m$ is homeomorphic to the ball $B_r(x)$ (this can be shown since $\mathbb{R}^m$ is homeomorphic to $B_1(0)$, and this ball is homeomorphic to any other by dilations and translations).

Now if $p \in M$ is any point and $V \subseteq M$ is a neighborhood of $p$ homeomorphic to $\mathbb{R}^m \approx B_r(x)$ where $x \in \mathbb{R}^m$, let $f: V \to B_r(x)$ be a homeomorphism. It must be that $f|_{V - \{p\}}$ is homeomorphic to $B_r(x) - \{y\}$ where $f(p) = y$, since the restriction of a homeomorphism is a homeomorphism to its image.