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Math Help - Compactness and Hausdorff Topology Question

  1. #1
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    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.
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  2. #2
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    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 ?
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  3. #3
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    hey could you please explain the answer to me a little bit. As i am having a little bit of difficult following it
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  4. #4
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    Quote Originally Posted by monkey.brains View Post
    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 Qs , we get an open set containing L.
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