Is every infinite compact space with no isolated points uncountable?

Question

I know that every nonempty Hausdorff compact space with no isolated points is uncountable, so I was wondering if we could substitute the nonempty Hausdorff part with it being infinite.

Answer 1

No. For instance, you could take a countably infinite set with the indiscrete topology.

You could consider that example to be cheating, as its $T_0$ quotient does have isolated points (so the space has points which are "isolated" from all points that they are topologically distinguishable from at all). For an example which is additionally $T_0$ (even $T_1$), you could take a countably infinite set with the cofinite topology.