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Math Help - help me in Prove

  1. #1
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    help me in Prove

    Prove

    1)  X is a regular space,  A compact set and  B closed set,
     A \bigcap B = {\O}
    Then there is open sets U and V exist, such that
     A\subseteq U, \ B \subseteq V,\ A \bigcap B = {\O}


    2)  X is compact and hausdorff space and  f : X \rightarrow X is continuous function
    prove there is a closed set non empty such that f(A)= A
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  2. #2
    Senior Member roninpro's Avatar
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    For part two, define a sequence of sets in the following way: A_0=X, A_1=f(A_0), A_2=f(A_1), A_3=f(A_2),\ldots. Since f:X\to X is continuous and takes a compact space to a Hausdorff space, the closed map lemma applies; we can say that each A_i is a closed set. Let A=\bigcap A_i. Note that A is closed and f(A)=A. To complete the proof, it necessary to show that A is nonempty. I can't see how to do that at the moment, so hopefully somebody else can fill in the gap.

    I hope that this gives you some ideas, at any rate.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nice rose View Post
    Prove

    1)  X is a regular space,  A compact set and  B closed set,
     A \bigcap B = {\O}
    Then there is open sets U and V exist, such that
     A\subseteq U, \ B \subseteq V,\ A \bigcap B = {\O}
    Oh come on!

    Hint:

    Spoiler:
    For each a\in A there are disjoint neighborhood U_a,V_a with a\in U_a and B\subseteq V_a. Cover A with the set of all the U_a's and procure a finite subcover. What then?


    2)  X is compact and hausdorff space and  f : X \rightarrow X is continuous function
    prove there is a closed set non empty such that f(A)= A
    Let's see some work

    Hint:

    Spoiler:


    What happens if K=\left\{x\in X:f(x)\ne x\right\}=X?
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