Intersection of Closures of Rationals and Irrationals is Reals

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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 Intersection is Idempotent.

$\blacksquare$


Sources