 Solution

An $n$-dimensional topological manifold is a Hausdorff, second countable topological space in which every point has an open neighbourhood homeomorphic to an open subset of ${\mathbb{R}}^n$. Equivalently, charts can be chosen onto open balls. This is the without-boundary convention.

A topological surface is a two-dimensional topological manifold. When connectedness is included in the convention for a surface, impose it as well; in the classification statement below the surface is explicitly connected. A closed surface means a compact surface without boundary, rather than merely a closed subset of another space.