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.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $30 \text { - } 31$. The Rational and Irrational Numbers: $3$