# Why can't that be an uncountable union?

I'm reading part of Lee's Introduction to manifolds. I have come to the following proposition.

$\textbf{Proposition 14.6 (Local Structure of Integral Manifolds).}$ Let $D$ be an involutive $k$-dimensional distribution on a smooth manifold $M$, and let $(U,\varphi)$ be a flat chart for $D$. If $N$ is any integral manifold of $D$, then $N\cap U$ is a countable disjoint union of open subsets of $k$-dimensional slices of $U$, each of which is open in $N$ and embedded in $M$.

Proof. Because the inclusion map $\eta:N\hookrightarrow M$ is continuous, $N\cap U=\eta^{-1}(U)$ is open in $N$, and thus consists of a countable disjoint union of connected components, each of which is open in $N$.

The proof then continues, and I will read the rest shortly. I was just wondering: why can't $\iota^{-1}(U)$ be an uncountable union of disjoint connected open components?