First Countable Spaces are Hausdorff or Not?

Question

Does first countable imply Hausdorff? If not, what is an example of first countable space that is not Hausdorff?

Answer 1

$\pi$-Base is a database of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following twenty-two first countable spaces that are not Hausdorff. You can learn more about any of them from the search result.

Compact Complement Topology

Countable Excluded Point Topology

Countable Particular Point Topology

Deleted Integer Topology

Divisor Topology

Double Pointed Reals

Either-Or Topology

Finite Complement Topology on a Countable Space

Finite Excluded Point Topology

Finite Particular Point Topology

Hjalmar Ekdal Topology

Indiscrete Topology

Interlocking Interval Topology

Nested Interval Topology

Odd-Even Topology

Overlapping Interval Topology

Prime Ideal Topology

Right Order Topology on $\mathbb{R}$

Sierpinski Space

Telophase Topology

Uncountable Excluded Point Topology

Uncountable Particular Point Topology

Answer 2

Consider any topological space with at least two points and the indiscrete topology: It is first countable but not Hausdorff.

Answer 3

As mathmax points out, first countability doesn’t imply even the weakest separation axiom, $T_0$. Moreover, adding some separation doesn’t help: first countability doesn’t imply Hausdorffness even for $T_1$ spaces, since the cofinite topology on $\Bbb N$ is first countable and $T_1$ but does not have any disjoint non-empty open sets.