# A non-Hausdorff space with a Hausdorff subspace [closed]

Can anyone give an example of a non-Hausdorff space that contains a Hausdorff subspace?

## closed as off-topic by TastyRomeo, projectilemotion, Namaste, JonMark Perry, Carsten SMar 11 '17 at 11:57

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• @KevinCarlson. I agree. Every text or paper I've read allows the empty set to be a top. space . It would be awkward if it weren't. It would be like allowing $1$ to be prime: Almost every mention of primes would have to be changed to "primes greater than $1$". – DanielWainfleet Mar 10 '17 at 21:50
• @AlessandroCodenotti: How about "the intersection of two subspaces is a subspace", or "every set can be given the discrete topology", or "the fiber over a point $b$ of a bundle $E \to B$" or "given a continuous map $X \to Y$, the inverse image of an open subspace is an open subspace". These basic notions all become very awkward when you exclude the empty space. – Hurkyl Mar 11 '17 at 3:14
• For those who don't recognize it from the description, add an additional point $0'$ to the real numbers and take the sets $(-\epsilon, 0)\cup\{0'\}\cup(0,\epsilon)$ to be a basis of neighborhoods of $0'$. $\Bbb R$ is a subspace, but $0$ and $0'$ are not separable. – Paul Sinclair Mar 11 '17 at 6:40