A surface is a topological space $S$ locally homeomorphic to $\mathbb{R}^2$, that is, for each point $p\in S$ there exists a neighborhood $U$ and a homeomorphism $\phi:U\longrightarrow V$ where $V\subset\mathbb{R}^2$ is open

(we may impose $V=\mathbb{R}^2$, it is enough to restrict to a ball centered in the target point, which is homeomorphic to the whole $\mathbb{R}^2$)