Folland 5.1.12 e) 
I'm working on part e) of the problem.
The quotient topology is defined as 
I'm reading the proof in http://www-users.math.umn.edu/~bahra004/for-real/real2-hw4.pdf.

But I'm still not sure why it suffices to show $\pi: X \rightarrow X / U$ is open. Also, I'm not sure what the topology defined by the quotient norm is.
Also, is there an alternative way of doing part e)?
Thanks!
 A: The quotient norm is the norm $\|\cdot\|_{\mathcal X/\mathcal M}: \mathcal {X/M}\to\Bbb R, \ x+\mathcal M \mapsto \inf_{y\in\mathcal M}\|x+y\|$.
The way the solution you have given works is as follows:
Let $\pi: A\to B$ be a surjective map and consider two topologies $\tau_1, \tau_2$ on $B$ so that $\pi$ is open and continuous wrt both $\tau_1,\tau_2$. Then $\tau_1=\tau_2$. The argument showing this is as follows:

Let $U\in\tau_1$ be open, then by continuity $\pi^{-1}(U)$ is open in $A$. By surjectivity you have that $U=\pi(\pi^{-1}(U))$ and by openness this is open in $\tau_2$, hence $\tau_1\subseteq \tau_2$. You may repeat the argument with $\tau_1$ and $\tau_2$ swapped.

In your case the quotient map to the quotient space topology is continuous by definition, the map to the quotient norm topology is continuous by c). The map to the quotient space topology is open since for $U\subseteq X$ open you have that:
$$\pi^{-1}[\pi(U)] = U+\mathcal M = \bigcup_{x\in\mathcal M}(U+\{x\})$$
which is open as a union of open sets, hence $\pi(U)$ is open by definition of the quotient space topology. The map to the quotient norm topology is open by the argument described in the solution.
