If $Z$ is Hausdorff and if $Z=M\oplus N$, then $M$ and $N$ are closed.

Question

Let $Z$ be a topological vector space over a field $K$.

If $Z$ is Hausdorff and if $Z=M\oplus N$, then $M$ and $N$ are closed.

Defn:

$X$ is said to be topological vector space if

$(i)$ $X$ is a vector space and the addition and scalar multiplications defined are continous.

$(ii)$ $(X,\tau)$ is a topological space.

Give me a hint to proceed with the proof...

Answer 1

The canonical projection $M\oplus N\to M$ is continuous and as $0\in M$ is closed ($M$ is Hausdorff, too), its preimage $N$ is closed.