Translation of Closed Set in Topological Vector Space is Closed Set

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 $x \in X$.

Then $F + x$ is a closed set in $X$.

Proof 1

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

We are given $F$ is closed in $X$.

So, $X \setminus F$ is open.

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

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

$X \setminus \paren {F + x} = \paren {X \setminus F} + x$

Since we have established that $\paren {X \setminus F} + x$ is open, we have that $X \setminus \paren {F + x}$ is open.

Hence by definition, $F + x$ is closed in $X$.

$\blacksquare$

Proof 2

Define a mapping $T_{-x} : X \to X$ by:

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

for each $y \in X$.

From Translation Mapping on Topological Vector Space is Homeomorphism, $T_{-x}$ is a homeomorphism.

From Definition 4 of a homeomorphism, $T_{-x}$ is therefore a closed mapping.

Hence $T_{-x} \sqbrk F = F + x$ is closed.

$\blacksquare$