About the classification of plane surfaces

ModusPonens

Well-known member
Hello.

Is it true that, in $\mathbb{R}^2$ the compact, connected 2-manifolds are determined by the number of holes in them? I think I saw this result somewhere, but I don't remember for sure.

If this is true would that mean that any compact connected set with non empty interior in $\mathbf{C}$ is homeomorphic to the closed unit ball (or disk)?

Thank you in advance.

Active member
Hello.

Is it true that, in $\mathbb{R}^2$ the compact, connected 2-manifolds are determined by the number of holes in them? I think I saw this result somewhere, but I don't remember for sure.

If this is true would that mean that any compact connected set with non empty interior in $\mathbf{C}$ is homeomorphic to the closed unit ball (or disk)?

Thank you in advance.
That first statement sounds correct, but I can't provide a citation or proof.

The second statement does not follow. For example, a compact punctured disk, e.g.
$$\left\{z\in \mathbb{C}:\,\frac{1}{2}\leq \left| z \right| \leq 1\right\}$$
Has a non-empty interior but is homeomorphic to the unit circle rather than the closed unit disk.