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**dwsmith** A subspace of a Hausdorff space is Hausdorff.

Let X be Hausdorff and $\displaystyle Y\subset X$

Since Y is a subspace, $\displaystyle Y=X\cap C$ where C is some open set in X.

Let $\displaystyle x_1,x_2\in X\cap C$

$\displaystyle X\cap C$ is the intersection of open sets so it is open. Therefore, $\displaystyle x_1\in U \ \text{and} \ x_2\in V$ where U and V are open sets in $\displaystyle X\cap C$. Again since $\displaystyle X\cap C$ is open, U and V can be constructed in such a manner that $\displaystyle U\cap V=\O$.

Thus, the subspace is Hausdorff.

Correct?