Dilation of Closure of Set in Topological Vector Space is Closure of Dilation

Theorem
Let $F$ be a topological field.

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

Let $A \subseteq X$.

Let $\lambda \in F$.

Then we have:


 * $\lambda A^- = \paren {\lambda A}^-$

where $A^-$ denotes the closure of $A$.

Proof
First suppose that $\lambda = 0$.

Then $\lambda A^- = \set { {\mathbf 0}_X}$ and $\lambda A = \set { {\mathbf 0}_X}$.

From Finite Topological Space is Compact, $\set { {\mathbf 0}_X}$ is a compact subspace of $X$.

Since $X$ is a topological vector space, it is Hausdorff, and so $\set { {\mathbf 0}_X}$ is closed from Compact Subspace of Hausdorff Space is Closed.

From Set is Closed iff Equals Topological Closure, we then have:


 * $\set { {\mathbf 0}_X}^- = \set { {\mathbf 0}_X}$

and so $\lambda A^- = \paren {\lambda A}^-$.

Now suppose that $\lambda \ne 0$.

From Set Closure as Intersection of Closed Sets, we have:


 * $\ds A^- = \bigcap \leftset {K \supseteq A : K}$ is closed in $\rightset X$

and:


 * $\ds \paren {\lambda A}^- = \bigcap \leftset {K \supseteq \lambda A : K}$ is closed in $\rightset X$

For brevity write:


 * $\ds \mathcal S_1 = \leftset {K \supseteq A : K}$ is closed in $\rightset X$

and:


 * $\ds \mathcal S_2 = \leftset {K \supseteq \lambda A : K}$ is closed in $\rightset X$

so that:


 * $\ds A^- = \bigcap_{K \in \mathcal S_1} K$

and:


 * $\ds \paren {\lambda A}^- = \bigcap_{K' \in \mathcal S_2} K'$

From Dilation of Intersection of Subsets of Vector Space, it now suffices to show that:


 * $\mathcal S_2 = \set {\lambda K : K \in \mathcal S_1}$.

Let $K \in \mathcal S_1$.

Then $K$ is closed and $A \subseteq K$.

From Dilation of Closed Set in Topological Vector Space is Closed Set, $\lambda K$ is closed.

Since $\lambda A \subseteq \lambda K$, we have:


 * $\set {\lambda K : K \in \mathcal S_1} \subseteq \mathcal S_2$

Now let $K' \in \mathcal S_2$.

Then $K'$ is closed and $\lambda A \subseteq K'$.

Let $K = \lambda^{-1} K'$.

From Dilation of Closed Set in Topological Vector Space is Closed Set, $K$ is closed.

We have $A \subseteq K$, so $K \in \mathcal S_1$.

Since we have $K' = \lambda K$ for $K \in \mathcal S_1$, we have:


 * $K' \in \set {\lambda K : K \in \mathcal S_1}$

giving:


 * $\mathcal S_2 \subseteq \set {\lambda K : K \in \mathcal S_1}$

So:


 * $\mathcal S_2 = \set {\lambda K : K \in \mathcal S_1}$

by the definition of set equality.

So: