Let A and B be disjoint compact subsets of a Hausdorff topological space X. Show that there exist disjoint open sets U and V containing A and B, respectively.
Let and be such compacts. Take one point in
For every point of there are two open sets and such that and and .
so for a finite subset of .
Now is an open set disjoint from (it is disjoint from ) which contains
Repeat that operation for every You obtain a set of open sets such that and is disjoint from (more precisely from ).
So same properties (and even better) for for a finite subset of
what can you say of and ?
First suppose that is a compact set and . Now prove that there two open sets, , such that .
That is the first part of the proof above.
Now in the second part of the proof we have two disjoint compact sets, .
Because they are disjoint, .
So apply the first part of the theorem. Because is compact we get a finite collections of open sets the union of which is an open set containing .
Intersecting the corresponding , we get an open set containing .