In Munkres Topology I've just finished the proof of the statement:
Theorem: Let X be a non-empty compact Hausdorff space with no isolated points. Then X is uncountable.
My question is that earlier in the text, when Hausdorff spaces were introduced, there was a theorem stating that all finite point sets in Hausdorff spaces were closed. So why do we need the added statement that there are no isolated points? Wouldn't $X$ being Hausdorff imply that there are no isolated points? Note that Munkres defines $x \in X$ to be isolated if $\{x\}$ is open in $X$.
No, being Hausdorff does not mean that there are no isolated points. As an example, just take the integers with the usual metric, that is, the induced topology is the discrete topology. Since it is a metric space, it must be Hausdorff. Also, all points here are isolated points.