First Countable Spaces are Hausdorff or Not? Does first countable imply Hausdorff? If not, what is an example of first countable space that is not Hausdorff?
 A: 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.
A: $\pi$-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
A: Consider any topological space with at least two points and the indiscrete topology: It is first countable but not Hausdorff.
