# 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)?

##### 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)?

$$\left\{z\in \mathbb{C}:\,\frac{1}{2}\leq \left| z \right| \leq 1\right\}$$