The following proof above is given in Introduction to Algebraic Topology by Rotman and I understand it except for the part where it's claimed that $y \in h(N) \setminus h(\dot{N})$. Now I understand that we must have $y \in h(N)$ since $h(x) = y$ and $x \in N$, but I don't see why $y \not\in h(\dot{N})$.
Why is it the case that $y \not\in h(\dot{N})$?
$x\in N$, $x\notin\dot N$. As $h$ is a homeomorphism, it is bijective, so $h(x)\in h(N)$ and $h(x)\notin h(\dot N)$.