Closure of Irrational Numbers is Real Numbers

Theorem

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

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

Then:

$\paren {\R \setminus \Q}^- = \R$

where $\paren {\R \setminus \Q}-$ denotes the closure of $\R \setminus \Q$.

Proof

From Irrationals are Everywhere Dense in Reals, $\R \setminus \Q$ is everywhere dense in $\R$.

It follows by definition of everywhere dense that $\paren {\R \setminus \Q}^- = \R$.

$\blacksquare$