# Connect the dots

Construct a countable, connected topological space with at least two points, satisfying the highest separability axiom you can.

Notice that if two points can be separated by a continuous function, then a connected space with more than one point is uncountable, so your space cannot be $T_{3 \frac 12}$ (or higher).

1. The Sierpiński space is a $T_0$ example, and this is as good as finite examples can get, since a finite $T_1$ space is discrete and therefore not connected if it has more than one point.