= 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.
Back to article page
