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Math Help - Quotient map from Hausdorff to non-Hausdorff space

  1. #1
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    Quotient map from Hausdorff to non-Hausdorff space

    Does a quotient map p : X to Y where X is Hausdorff and Y is not exist?
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  2. #2
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    Quote Originally Posted by claves View Post
    Does a quotient map p : X to Y where X is Hausdorff and Y is not exist?
    Choose a quotient map p:\mathbb{Re}_K \rightarrow Y, where \mathbb{Re}_K denotes the real line in the K-topology and Y be the quotient space obtained from \mathbb{Re}_K by collapsing the set K to a point.

    \mathbb{Re}_K is Hausdorff, but Y is not Hausdorff since p(K) and p(0) in Y cannot be separated by disjoint open sets.
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