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Math Help - Continuity, dense and Hausdorff

  1. #1
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    Continuity, dense and Hausdorff

    Let f,g: X->Y be a continuous function. Assume that Y is Hausdorff and that there exists a dense subset D of X such that f(x)=g(x) for all x \in D .

    Prove that f(x)=g(x) for all x \in X
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Let f,g: X->Y be a continuous function. Assume that Y is Hausdorff and that there exists a dense subset D of X such that f(x)=g(x) for all x \in D .

    Prove that f(x)=g(x) for all x \in X
    Since f(x)=g(x) for all x \in D by the hypothesis of the problem, it remains to show that f(x)=g(x) for all  x \in X \setminus D.

    Suppose to the contrary that f(x) \neq g(x) for some  x \in X \setminus D. Since Y is Hausdorff, we have disjoint open sets U and V containing f(x) and g(x), respectively. Since f and g are continuous, x belongs to an open set f^{-1}(U) \cap g^{-1}(V). We know that D is dense in X, which implies that every open set in X intersects a dense subset D. Thus, an open set f^{-1}(U) \cap g^{-1}(V) intersects D. Let y belongs to that intersection. Since y belongs to f^{-1}(U) \cap g^{-1}(V) , f(y) \neq g(y). Since y also belongs to D, we have f(y)=g(y) by the hypothesis of the problem. Contradiction !

    Thus, f(x) = g(x) for all  x \in X \setminus D.
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