Dilation of Closed Set in Topological Vector Space is Closed Set/Proof 1

Theorem

Let $K$ be a topological field.

Let $X$ be a topological vector space over $K$.

Let $F$ be a closed set in $X$.

Let $\lambda \in K \setminus \set {0_K}$.

Then $\lambda F$ is a closed set in $X$.

Proof

We aim to show that $X \setminus \paren {\lambda F}$ is open.

Since $F$ is closed, $X \setminus F$ is open.

It follows from Dilation of Open Set in Topological Vector Space is Open that $\lambda \paren {X \setminus F}$ is open.

From Dilation of Complement of Set in Vector Space, we have:

$X \setminus \paren {\lambda F} = \lambda \paren {X \setminus F}$.

Since we have established that $\lambda \paren {X \setminus F}$ is open, it follows that $X \setminus \paren {\lambda F}$ is open.

So $\lambda F$ is closed.

$\blacksquare$