Set of Condensation Points of Countable Set is Empty/Lemma

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $A$ be a subset of $S$.

Then:
 * if $A$ is countable,
 * then there exists no point $x$ of $S$ such that $x$ is a condensation point of $A$.

Proof
Assume
 * $A$ is countable.

there exists a point $x$ of $S$ such that
 * $x$ is a condensation point of $A$

By definition of topological space:
 * $S \in \tau$

Then by definition of condensation point:
 * $A \cap S$ is uncountable

By Intersection with Subset is Subset:
 * $A \cap S = A$

$A$ is countable contradicts $A$ is uncountable.

Thus the result follows by Proof by Contradiction.