Dilation of Closed Set in Topological Vector Space is Closed Set/Proof 2

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

Define a mapping $c_\lambda : X \to X$ by:

$\map {c_\lambda} x = \lambda x$

for each $x \in X$.

From Dilation Mapping on Topological Vector Space is Homeomorphism, $c_\lambda$ is a homeomorphism.

From Definition 4 of a homeomorphism, $c_\lambda$ is therefore a closed mapping.

Hence $c_\lambda \sqbrk F = \lambda F$ is closed.

$\blacksquare$