# Space which is $T_1$ and sober but not Hausdorff?

Every Hausdorff space is $$T_1$$ and sober. Does the converse hold? I expect not. What's a counterexample?

I expected I should be able to look this up in Counterexamples in Topology, but unfortunately that book doesn't appear to discuss sober spaces.

• Wikipedia has an example. "Let X be the set of real numbers, with a new point p adjoined; the open sets being all real open sets, and all cofinite sets containing p." – Nate Eldredge Sep 20 '18 at 19:18
• @NateEldredge Thanks, I can't believe I missed that. – Tim Campion Sep 20 '18 at 19:22
• I suppose more generally one can take any infinite $T_1$ and sober space $Y$ and adjoin a new point in a similar manner to get a $T_1$ and sober space which is not Hausdorff. – Tim Campion Sep 20 '18 at 19:26
• I've just added references to this example to a few relevant nlab pages. – Tim Campion Sep 20 '18 at 19:42
• I recommend making Nate's comment a CW answer, accepting it, and then this question is resolved. – David White Sep 20 '18 at 20:31