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I would like to prove that if $X$ is, let's say, a normed vector space, and $A$ is a dense subset of $X$, and $x_0 \in X$, that $\{x_0\} + A$ is also dense in $X$.

I know that translation by a point is a homeomorphism, but not translation by a set. I have also tried using sequences, and using closure arguments. Is there any easy way to do this problem?

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A set is dense if it meets every non-empty open set. $V$ meets $x_0+A$ iff $V-x_0$ meets $A$.

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  • $\begingroup$ Of course. Thanks! $\endgroup$ – Johnny Apple Sep 23 '15 at 2:52
  • $\begingroup$ @Johnny: You're welcome! $\endgroup$ – Brian M. Scott Sep 23 '15 at 3:00
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An elementary proof, using sequences, is the following:

Let $x\in X$.

$A$ is dense and therefore there exists a sequence $\{x_n\}$ in $A$ such that $ x_n\to x-x_0$. So the sequence $y_n=x_0 + x_n$ is in $\left \{ x_0 \right \}+A$ and $y_n\to x_0+x-x_0=x$. So $x\in\overline{ \left \{ x_0 \right \}+A}$ and therefore $\left \{ x_0 \right \}+A$ is dense.

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