$A$ and $B$ disjoint and closed implies there is positive distance between both sets

Question

I recently came across this question: A and B disjoint, A compact, and B closed implies there is positive distance between both sets

Do a and b always have a positive distance for a belonging to A and b belonging to B?

I was wondering if this theory also applies if both A and B were closed? Both of them both part of a metric space.

My intuition says no but I don't know how to prove it.

are they inside an arbitrary metric space? in $\mathbb R^n$? What is the enviorment? — Asinomás, Dec 19 '16 at 07:09

score 1 · Answer 1 · answered Dec 19 '16 at 07:17

The answer for arbitrary metric spaces is no, for example consider the space $\mathbb R\setminus {0}$ and take the clopen sets $\mathbb R^+$ and $\mathbb R^-$.

The answer for real or complex normed vector spaces of finite dimension is yes, because closed balls are compact ( consequence for the equivalence of norm theorem). The answer for arbitrary vector spaces is no, finally it is true for convex sets inside Hilbert spaces.

$A$ and $B$ disjoint and closed implies there is positive distance between both sets

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