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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
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Trivial Topology –  Amitesh Datta Nov 16 '11 at 6:43

2 Answers 2

$\pi$-Base, an online version of the General Reference Chart from Steen and Seebach's Counterexamples in Topology, gives the following examples of connected, first countable spaces that are not Hausdorff. You can view the search result to learn more about these spaces.

Compact Complement Topology

Countable Excluded Point Topology

Countable Particular Point Topology

Divisor Topology

Finite Complement Topology on a Countable Space

Finite Excluded Point Topology

Finite Particular Point Topology

Interlocking Interval Topology

Nested Interval Topology

Overlapping Interval Topology

Prime Ideal Topology

Right Order Topology on R

Sierpinski Space

Telophase Topology

Uncountable Excluded Point Topology

Uncountable Particular Point Topology

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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.

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