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

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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?

Answer 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 $0 <r <\frac {d(A,B)} 2$.