The classification of surfaces

Question

Can we completely classify the simply-connected surfaces (with or without boundary) in $\mathbb R^3$ up to homeomorphism?

Answer 1

The classification of surfaces is well-known. Every surface is homeomorphic to either $mT+nD$ or $mP+nD$ where $T$ is a torus, $P$ is a projective plane, $D$ is a disk, $m,n$ are non-negative integers, and $+$ is connected sum.

Now you want the surfaces to be in ${\bf R}^3$, so that rules out $mP$ for $m\ge1$.

You also want them simply-connected. That rules out $mT+nD$ for $m\ge1$ and $mP+nD$ for $m\ge1$ and $nD$ for $n\ge2$, leaving only two surfaces: the sphere, and the disk.