I'm studying how to work with surfaces in differential geometry. The definition of a regular surface is the following one:
A subset $S\subset\mathbb{R}^3$ is a regular surface if, for each $p\in S$, there exist a neighborhood $V\subset\mathbb{R}^3$ and a map $x:U\to V\cap S$ where $U$ is an open subset of $\mathbb{R}^2$ such that $x$ is differentiable ($C^\infty$), an homeomorphism and its differential, $dx_q:\mathbb{R}^2\to\mathbb{R}^3$ injective.
My question is, each time I pick a point $p\in S\cap V$, and take its preimage, $x^{-1}(p)$, the set $x^{-1}(p)$ always consists in only one point? (since homeomorphisms are injective).