Definition of a "Surface" -- an open ball or an open subset of $R^2$?

Question

I have learnt that surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane $R^2$. --- (a)

But then, it seems that people use another definition that "a surface is ... that has a neighborhood homeomorphic to an open ball in $R^2$". --- (b)

Clearly, (b) implies (a), but does (a) also imply (b)?

Answer 1

Yes, it does. Let $p$ be the point on the surface and $\varphi$ the homeomorphism. The open subset of $\mathbb R^2$ contains an open ball around each point by definition. Take an open ball around $\varphi(p)$. This ball is homeomorphic to an open neighborhood of $p$ via $\varphi^{-1}$ restricted to the ball.