Translation of Closed Set in Topological Vector Space is Closed Set/Proof 1
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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
We aim to show that $X \setminus \paren {F + x}$ is open.
Since $F$ is closed, $X \setminus F$ 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.
So $F + x$ is closed.
$\blacksquare$