Uncountable Subset of Countable Complement Space Intersects Open Sets

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Let $T = \struct {S, \tau}$ be a countable complement topology on an uncountable set $S$.

Let $H \subseteq S$ be an uncountable subset of $S$.

Then the intersection of $H$ with any non-empty open set of $T$ is uncountable.


Let $U \in \tau$ be any non-empty open set of $T$.

Then $\relcomp S U$ is countable.

Suppose $H \cap U = \O$.

Then from Intersection with Complement is Empty iff Subset it follows that $H \subseteq \relcomp S U$ and so $H$ is countable.

So if $H$ is uncountable it is bound to have a non-empty intersection with every open set in $T$.