Surface

Remarks

Example: the unit sphere $S^2 \subset \R^3$ is a surface. It is Hausdorff and second countable as a subspace of $\R^3$. Stereographic projections from the north pole and from the south pole are homeomorphisms of two open subsets of $S^2$ onto $\R^2$, and these subsets cover $S^2$.

Non-example: let $X$ be the union of the planes $z = 0$ and $y = 0$ in $\R^3$. Clauses 1 and 2 hold, but clause 3 fails at points of the $x$-axis. A small sphere around a point $P$ of this axis meets $X$ in two great circles crossing at two points, a graph $G$ with $2$ vertices and $4$ edges. So $P$ has a neighbourhood that is a cone on $G$, and the local homology group $H_2(X, X \setminus \set{P})$ is $\tilde H_1(G) \cong \Z^3$. At a point of an open subset of $\R^2$ this group is $\tilde H_1(S^1) \cong \Z$, and local homology is preserved by homeomorphisms of neighbourhoods.