Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
This criterion may come in handy –  Hagen von Eitzen Jan 6 '13 at 13:25
add comment

1 Answer 1

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.