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Math Help - Homeomorphism and metrizability

  1. #1
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    Homeomorphism and metrizability

    Prove that if X is a metrizable topological space and Y is homeomorphic to X, then Y is metrizable
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Prove that if X is a metrizable topological space and Y is homeomorphic to X, then Y is metrizable
    Since X is a metrizable topological space, we have a metric space (X, d).
    Let Y be a topological space homeomorphic to X and f:X \rightarrow Y be a homeomorphism.
    Define d' on Y \times Y such that

    d'(y_1, y_2) = d( f^{-1}(y_1), f^{-1}(y_2)), y_1, y_2 \in Y.

    I'll leave it to check d' is indeed a metric.

    Since both f and f^{-1} are continuous bijection, we see that f and f^{-1} are isometries, which implies that an open ball of radius r >0 with respect to a metric d on space X corresponds to an open ball of radius r with respect to a metric d' on space Y, and vice versa. Now, the open balls in Y defined by d' can be given as a basis for a topological space Y. Thus, Y is metrizable.
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