Irrational Number Space is Separable

Theorem

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.

Then $\struct {\R \setminus \Q, \tau_d}$ is separable.

Proof

Let $S$ be the set defined as:

$S = \set {\pi + q: q \in \Q}$

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

Therefore $S$ is also countable.

From $\pi$ is Irrational:

$\pi \in \R \setminus \Q$

It follows from Rational Addition is Closed that:

$\forall q \in \Q: \pi + q \in \R \setminus \Q$

and so:

$S \subseteq \R \setminus \Q$

From Rationals plus Irrational are Everywhere Dense in Irrationals, it follows that $S$ is everywhere dense in $\R \setminus \Q$.

Thus we have constructed a countable subset $S$ of $\R \setminus \Q$ which is everywhere dense in $\R \setminus \Q$.

The result follows by definition of separable space.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $31$. The Irrational Numbers: $7$