Uncountable Subset of Countable Complement Space Intersects Open Sets

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Theorem

Let $T = \left({S, \tau}\right)$ 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.


Proof

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

Then $\complement_S \left({U}\right)$ is countable.

Suppose $H \cap U = \varnothing$.

Then from Intersection with Complement is Empty iff Subset it follows that $H \subseteq \complement_S \left({U}\right)$ and so $H$ is countable.

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



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