The setting:
- $X$ and $Y$ are topological vector spaces.
- $N \subset X$ is a closed subspace.
- $T(N)=\{0\}$
- $\pi : X \rightarrow X/N$ the quotient map.
- $S : X/N \rightarrow Y$ uniquely determined by $T=S \circ \pi$ ($S$ is well-defined by $S(x+N)=T(x)$ and is linear).
The homework:
- $T$ is continuous iff $S$ is continuous.
- $T$ is open iff $S$ is open.
I "proved" both statements, but I guess I have a mistake because I never used the fact that $N$ is closed.
My attempt:
- If $T$ is continuous and $U \subset Y$ is open than $\pi^{-1}(S^{-1}(U))=T^{-1}(U)$ is open because $T$ is continous.
- I showed that $\pi$ is open and thus if $S$ is open, so is $T=S \circ \pi$. As for the other direction, I showed that if $B\subset X/N$ is open then $S(B)=T(A)$ for some $A \subset X$ such that $A+N$ is open. So $S(B)=T(A)=T(A+N)$ is open if $T$ is open.
Is it really necessary that $N$ is closed?