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.

Aiming for a contradiction, suppose 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.

$\blacksquare$

Sources

• 1989: Ryszard Engelking: General Topology (revised and completed ed.)

• Mizar article TOPGEN_4:55