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Math Help - Proof with closed sets....

  1. #1
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    Proof with closed sets....

    Suppose f:[a,b]--> R and g:[a,b]-->R. Let T={x:f(x)=g(x)}
    Prove that T is closed.

    So we can set h(x)=f(x)-g(x) and then T is the set such that h(x)=0 and so T complement is the set with h(x)=/=0 and then if we can show that this is open, then it's complement, T is closed. I don't know how to show it is open though. We need to show there is some neighborhood, but I don't know what to do. Thanks.
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  2. #2
    Moo
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    Hello,

    If f and g are continuous, then the proof is trivial, since for any continuous function h, h^{-1}(\text{a closed sed}) is a closed set.

    And here, you have h^{-1}(\{0\})...
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