Countable Set may have Uncountable Limit Points

Theorem

Let $S$ be an uncountable set.

Let $H \subseteq S$ be a countable subset of $S$.

Let $T = \struct {H, \tau}$ be a topological space on the underlying set $H$.

Then despite the fact that $H$ is countable, the set of limit points of $T$ may be uncountable.

Proof

Let $\R$ be the set of real numbers.

Let $\Q$ be the set of rational numbers.

Let $x \in \R$.

Then from Real Number is Limit Point of Rational Numbers in Real Numbers, $x$ is a limit point of $T$.

As $x$ is arbitrary, it follows that every element of $\R$ is a limit point of $T$.

From Rational Numbers are Countably Infinite, $\Q$ is a countable set.

From Real Numbers are Uncountable, $\R$ is an uncountable set.

Hence the result.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.10 \ \text {(c)}$