Translation of Closed Set in Topological Vector Space is Closed Set/Proof 2
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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
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.
$\blacksquare$