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

$\displaystyle \Rightarrow$

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

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