Compact Hausdorff Space with no Isolated Points is Uncountable/Lemma
Let $x \in S$.
First we show that there is a $y \in U$ such that $y \ne x$:
By Law of Excluded Middle, either $x \in U$ or $x \notin U$.
Let $x \in U$.
We have that $x$ is not an isolated point.
- $\exists y \in U: y \ne x$
Let $x \notin U$.
As $U$ is non-empty it has an element $y$, and $y \ne x$.
Thus in either case, there is a $y \in U$ such that $y \ne x$.
Law of the Excluded Middle
This proof depends on the Law of the Excluded Middle.