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Math Help - Hausdorff space

  1. #1
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    Hausdorff space

    Show that X is a Hausdorff space iff. the diagonal \Delta=\{x\times x:x\in X\} is a closed set in the product space.

    \Rightarrow
    Suppose X is a Hausdorff space. Then \forall x_1,x_2\in X such that x_1\neq x_2 there exists neighborhoods U_1 \ \text{and} \ U_2 of  x_1, x_2, respectively, such that U_1\cap U_2=\O.

    I don't know how I am supposed to use this to conclude the other piece.
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    Re: Hausdorff space

    Quote Originally Posted by dwsmith View Post
    Show that X is a Hausdorff space iff. the diagonal \Delta=\{x\times x:x\in X\} is a closed set in the product space.
    If (x,y) is such that x\ne y can you show that there is a ball centered at (x,y) that contains no point of \Delta.
    Hint: \frac{|x-y|}{4}>0.
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    Re: Hausdorff space

    Quote Originally Posted by Plato View Post
    If (x,y) is such that x\ne y can you show that there is a ball centered at (x,y) that contains no point of \Delta.
    Hint: \frac{|x-y|}{4}>0.
    I don't think I can use that. I don't know it is a Metric space.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Re: Hausdorff space

    Quote Originally Posted by dwsmith View Post
    Show that X is a Hausdorff space iff. the diagonal \Delta=\{x\times x:x\in X\} is a closed set in the product space.

    \Rightarrow
    Suppose X is a Hausdorff space. Then \forall x_1,x_2\in X such that x_1\neq x_2 there exists neighborhoods U_1 \ \text{and} \ U_2 of  x_1, x_2, respectively, such that U_1\cap U_2=\O.

    I don't know how I am supposed to use this to conclude the other piece.
    Merely note that if \Delta_X is closed, then X^2-\Delta_X is open, and so if x\ne y\in X then (x,y)\in X^2-\DeltaX and so there exists some basic open neighbohood U\times V of (x,y) with U\times V\subseteq X^2-\Delta_X. Conclude from this that x\in U, y\in V, and U\cap V=\varnothing.

    Now, try to reverse this to get the reverse implication you asked about.
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