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.

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- Apr 8th 2009, 09:50 AMr2dee6Compactness and Hausdorff Topology Question
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.

- Apr 8th 2009, 10:27 AMclic-clac
Hi

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 ? - Apr 21st 2009, 11:22 AMmonkey.brains
hey could you please explain the answer to me a little bit. As i am having a little bit of difficult following it

- Apr 21st 2009, 12:26 PMPlato
This is a standard two-part proof of this theorem.

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 .