Why can we cover a surface in $\Bbb{R}^3$ with countably many open orientable connected subsets?
This is a claim in my Differential Geometry book (Montiel and Ros, Curves and Surfaces, 2nd edition). To argue the authors invoke Lemma 3.2 below:
Lemma 3.2: Let $S$ be a surface and let $X: U \longrightarrow \Bbb{R}^3$ be a parametrization of $S$. Then there is a unit normal field defined on the open set $V = X(U)$.
Now, as I understand, Lemma 3.2 gives us a cover of $S$ with orientable connected subsets. Why can there be only countably many? What am I missing?
Thanks in advance and kind regards.