Irrational Numbers form G-Delta Set in Reals

Theorem

Let $\R \setminus \Q$ denote the set of irrational numbers.

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

Then $\R \setminus \Q$ forms a $G_\delta$ set in $\R$.

Proof

\(\ds \Q\)

\(=\)

\(\ds \bigcup_{\alpha \mathop \in \Q} \set \alpha\)

Rational Numbers form F-Sigma Set in Reals

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\(\ds \leadsto \ \ \)

\(\ds \R \setminus \Q\)

\(=\)

\(\ds \R \setminus \bigcup_{\alpha \mathop \in \Q} \set \alpha\)

\(\ds \)

\(=\)

\(\ds \bigcap_{\alpha \mathop \in \Q} \paren {\R \setminus \set \alpha}\)

De Morgan's Laws: Difference with Union

The result follows from Rational Numbers are Countably Infinite.

$\blacksquare$

Sources

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