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Let $\bar D $ and $\bar E$ denote the closures of $D$ and $E$ respectively. If $ D\subset \mathbb R^n$, $E \subset \mathbb R^n$ and they are strongly separated. Show that $\bar D $ and $\bar E$ can also be separated strongly.

I got stuck with this proof. I want to use the strong separation proof but I don't know which step to take. Thank you for your help.

Stefan Hansen
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GeorgeMasonn
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1 Answers1

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Defining $d(A,B) = \inf_{a \in A, b \in B} d(a,b)$, being strongly separated is the condition $d(A,B) > 0$.

It suffices to observe that $$d(A,B) = d(\bar{A}, \bar{B})$$

Indeed, $\geqslant$ is immediate, for $\leqslant$, if $\bar{a} \in \bar{A}, \bar{b} \in \bar{B}$, then by taking sequences $a_i$, $b_i$ approaching $\bar{a}$ and $\bar{b}$, we have $$\lim_i d(a_i, b_i) = d(\bar{a}, \bar{b})$$by the triangle inequality.

uncookedfalcon
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