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.
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.
$\pi$-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
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.