Intersection of Closures of Rationals and Irrationals is Reals

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

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

Then:
 * $\Q^- \cap \paren {\R \setminus \Q}^- = \R$

where:
 * $\R \setminus \Q$ denotes the set of irrational numbers
 * $\Q^-$ denotes the closure of $\Q$.

Proof
From Closure of Rational Numbers is Real Numbers:
 * $\Q^- = \R$

From Closure of Irrational Numbers is Real Numbers:
 * $\paren {\R \setminus \Q}^- = \R$

The result follows from Set Intersection is Idempotent.