Rational Numbers are F-Sigma Set in Real Line

Theorem

Let $\struct {\R, \tau}$ be the real number line considered asa topological space with the usual (Euclidean) topology.

Then:

$\Q$ is an $F_\sigma$ set in $\struct {\R, \tau}$.

Proof

Define the set of subsets of $\R$ as:

$\FF := \set {\set x: x \in \Q}$

By Closed Real Interval is Closed Set:

$\forall x \in \Q: \closedint x x = \set x$ is closed (in topological sense)

Then:

$\forall A \in \FF: A$ is closed

By Cardinality of Set of Singletons:

$\card \FF = \card \Q$

where $\card \FF$ denotes the cardinality of $\FF$.

By Rational Numbers are Countably Infinite:

$\Q$ is countable.

Therefore by Set is Countable if Cardinality equals Cardinality of Countable Set:

$\FF$ is countable.

By Union of Set of Singletons:

$\bigcup \FF = \Q$

Thus, by definition, $\Q$ is an $F_\sigma$ set.

$\blacksquare$

Sources

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

• Mizar article TOPGEN_4:45