Existence of disjoint open sets $U$ and $V$ which are supersets of disjoint compact sets $A$ and $B$

Question

I am not familiar with Hausdorff spaces so I need to derive the above result using my existing knowledge which is:

In the context of metric spaces only, I already know that if two compact sets $A$ and $B$ are disjoint, then the distance between them is strictly positive.

Given this knowledge, is it possible to prove $A \subset U$ and $B \subset V$ for $U, V$ open and disjoint? How?

Metric spaces are Hausdorff... Learning what Hausdorff spaces are is worth your time. — Rushabh Mehta, Feb 17 '20 at 04:23
For the metric space case the proof is very easy: simply define $U={x: d(x,A) <r}$ and $V={x: d(x,B) <r}$ where $0<r<d(A,B)$. The general case is slightly lengthier. — Kavi Rama Murthy, Feb 17 '20 at 05:41
@KaviRamaMurthy If you want, you can put this as an answer and I will accept it. btw should it be $0 < r \leq \frac{d(A, B)}{2}$ ? — rims, Feb 17 '20 at 16:06

score 1 · Accepted Answer · answered Feb 17 '20 at 23:13

1

In the case of a metric space you can just take $U=\{x:d(x,A) <r\}$ and $V=\{x:d(x,B) <r\}$ where $0 <r <\frac {d(A,B)} 2$.

answered Feb 17 '20 at 23:13

Kavi Rama Murthy

359,332

Existence of disjoint open sets $U$ and $V$ which are supersets of disjoint compact sets $A$ and $B$

1 Answers1