Closure of Rational Numbers is Real Numbers

Theorem

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

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

Then:

$\Q^- = \R$

where $\Q^-$ denotes the closure of $\Q$.

Proof

From Rationals are Everywhere Dense in Topological Space of Reals, $\Q$ is everywhere dense in $\R$.

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

$\blacksquare$