# Is a connected, first countable space necessarily Hausdorff?

I've been trying for forever to come up with a counterexample but haven't had any luck.

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We do not remove questions just because they have been answered. To show that you are satisfied with an answer, click the checkmark next to the answer you are accepting. –  Austin Mohr Nov 16 '11 at 6:21
Trivial Topology –  Amitesh Datta Nov 16 '11 at 6:43
The Sierpinski Two-Point Space is connected and first countable, but not Hausdorff (indeed, not even $T_1$).
Another example: the cofinite topology on the integers is $T_1$, second countable (so first countable), connected and very non-Hausdorff for the same reason: all non-empty open sets intersect.