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Math Help - Continuity from Restriction Continuities

  1. #1
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    Continuity from Restriction Continuities

    Let X, Y be topological spaces, and A,B be subsets of X such that X = AUB. Let f:X --> Y be a map such that f restricted to A and f restricted to B are continuous maps w.r.t. the subspace topology. Show:

    (a) If A and B are closed, f is continuous.
    (b) If A and B are open, is f always continuous? Prove or give a counterexample.
    (c) If A is open and B is closed, is f always continuous? Prove or give a counterexample.

    I think both (a) and (b) are true, but I'm not quite sure how to attack it. To show continuity we are basically trying to show that the pre-image of an open set is open (or equivalently the pre-image of a closed set is closed), but I'm not quite sure how to use these hypotheses here.

    I think (c) is false, I can construct a closed set with an open set inside of it that will just have different constant values on those sets.
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  2. #2
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    Quote Originally Posted by joeyjoejoe View Post
    Let X, Y be topological spaces, and A,B be subsets of X such that X = AUB. Let f:X --> Y be a map such that f restricted to A and f restricted to B are continuous maps w.r.t. the subspace topology. Show:

    (a) If A and B are closed, f is continuous.
    (b) If A and B are open, is f always continuous? Prove or give a counterexample.
    (c) If A is open and B is closed, is f always continuous? Prove or give a counterexample.

    I think both (a) and (b) are true, but I'm not quite sure how to attack it. To show continuity we are basically trying to show that the pre-image of an open set is open (or equivalently the pre-image of a closed set is closed), but I'm not quite sure how to use these hypotheses here.

    I think (c) is false, I can construct a closed set with an open set inside of it that will just have different constant values on those sets.
    For (a), suppose that G is a closed subset of Y. Then f^{-1}(G) = f^{-1}(G)\cap(A\cup B) = (f^{-1}(G)\cap A)\cup(f^{-1}(G)\cap B). That is the union of two closed sets and is therefore closed.

    For (b), the argument is the same, with "closed" replaced throughout by "open".
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