If $N$ is a submanifold of $M$ and $V$ is a connected open subset of $M$, then $N\cap V$ is the union of a countable colection of connected open subsets of $N$.
This is a exercise from Boothby, An introduction to differential manifolds and Riemann Geometry, p. 81.
I can't see a clearly way to start this, anyone can give me some steps?
The basic start is taking the injective inclusion maps to $N$ from $M$ since $N$ is a submanifolds, but then how should I proceed? Taking local charts?
Here's my (potentially naive) thought process for this problem, and I hope that it can be of some help!
My first thought was that we want to show $N\cap V=\bigcup_{j\geq 1}C_j$, where $C_j$ is connected and open in $N$, i.e. $C_j=N\cap W_j$ for $W_j$ open in $M$ and $C_j$ is contained in a connected component of $N$.
Let $\mathcal{B}=\{B_i\}_{i\geq 1}$ be a countable basis for $N$. Note as each $B_i$ is homeomorphic to an open ball in Euclidean space, they are also connected (in fact, path connected). We then see by the distributive laws of set theory that
$N\cap V = \left(\bigcup_{i\geq 1} B_i \right)\cap V = \bigcup_{i\geq 1}\left( B_i\cap V\right),$
where it can be shown that $B_i\cap V$ is a countable collection of connected open sets in $N$. Taking the union over $i$ then gives a countable collection of open and connected subsets of $N$, as desired.
Feel free to critique/point out any holes in my thought process since I'm sure some exist! (They always seem to.)
EDIT: fixed some false statements and fixed some holes in the argument.