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 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: $3$