Compactness and Hausdorff Topology Question

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April 8th 2009, 09:50 AM
r2dee6

Compactness 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.
April 8th 2009, 10:27 AM
clic-clac

Hi

Let $K$ and $L$ be such compacts. Take one point $k$ in $K.$

For every point $l$ of $L,$ there are two open sets $U_l$ and $V_l$ such that $k\in U_l$ and $l\in V_l$ and $U_l\cap V_l=\emptyset$ .

$K\subseteq \bigcup\limits_{l\in L}V_l$ so $K\subseteq \bigcup\limits_{l\in I_k}V_l$ for a finite subset $I_k$ of $L$ .

Now $O_k:=\bigcap\limits_{l\in I_k}U_l$ is an open set disjoint from $L$ (it is disjoint from $\bigcup\limits_{l\in I_k}V_l$ ) which contains $k .$

Repeat that operation for every $k\in K .$ You obtain a set of open sets $\{O_k;\ k\in K\}$ such that $K\subseteq\bigcup\limits_{k\in K}O_k$ and $\bigcup\limits_{k\in K}O_k$ is disjoint from $L$ (more precisely from $\bigcap\limits_{k\in K}\bigcup\limits_{l\in I_k}V_l$ ).

So same properties (and even better) for $\bigcup\limits_{k\in J}O_k$ for a finite subset $J$ of $K :$

what can you say of $\bigcap\limits_{k\in J}\bigcup\limits_{l\in I_k}V_l$ and $\bigcup\limits_{k\in J}O_k$ ?
April 21st 2009, 11:22 AM
monkey.brains

hey could you please explain the answer to me a little bit. As i am having a little bit of difficult following it
April 21st 2009, 12:26 PM
Plato

Quote:

Originally Posted by monkey.brains

hey could you please explain the answer to me a little bit. As i am having a little bit of difficult following it

This is a standard two-part proof of this theorem.
First suppose that $L$ is a compact set and $p \notin L$ . Now prove that there two open sets, $O~\&~Q$ , such that $p \in O,~L \subseteq Q\,\& \,O \cap Q = \emptyset$ .
That is the first part of the proof above.

Now in the second part of the proof we have two disjoint compact sets, $K~\&~L$ .
Because they are disjoint, $\left( {\forall p \in K} \right) \Rightarrow \quad p \notin L$ .
So apply the first part of the theorem. Because $K$ is compact we get a finite collections of open sets the union of which is an open set containing $K$ .
Intersecting the corresponding $Q’s$ , we get an open set containing $L$ .