Set of Rational Numbers is not G-Delta Set in Reals
Theorem
Let $\Q$ be the set of rational numbers.
Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.
Then $\Q$ is not a $G_\delta$ set in $\R$.
Proof
Aiming for a contradiction, suppose $\Q$ is a $G_\delta$ set in $\R$.
Let $\Q = \ds \bigcap_{i \mathop \in \N} V_i$.
Since Rational Numbers are Countably Infinite, there exists an enumeration of $\Q$.
Write $\Q = \set {q_i}_{i \mathop \in \N}$.
Define $U_i = V_i \setminus \set {q_i}$.
We show that $U_i$ is dense in $\R$.
- Let $A \subseteq \R$ be an open set of $\struct {\R, \tau}$.
- From Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\R$ form a basis for $\struct {\R, \tau}$.
- So there exists $\openint a b \subseteq A$ for some $a < b$.
- One of $q_i \le a$, $q_i > a$ must hold.
- Suppose $q_i \le a$.
- By Between two Real Numbers exists Rational Number we have:
- $\exists r \in Q: a < r < b$.
- Since $r > a \ge q_i$:
- $r \in \Q \setminus \set {q_i} \subseteq V_i \setminus \set {q_i} = U_i$.
- Suppose $q_i > a$.
- By Between two Real Numbers exists Rational Number we have:
- $\exists r \in Q: a < r < \min \set {q_i, b}$.
- Since $r < q_i$:
- $r \in \Q \setminus \set {q_i} \subseteq V_i \setminus \set {q_i} = U_i$.
- In both cases we see that $r \in \openint a b \cap U_i \subseteq A \cap U_i$.
- Therefore $A \cap U_i \ne \O$.
- Thus $U_i$ is dense in $\R$.
Also we have:
\(\ds \bigcap_{i \mathop \in \N} U_i\) | \(=\) | \(\ds \bigcap_{i \mathop \in \N} \paren {V_i \setminus \set {q_i} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\bigcap_{i \mathop \in \N} V_i} \setminus \paren {\bigcup_{i \mathop \in \N} \set {q_i} }\) | Difference with Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \Q \setminus \Q\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \O\) | Set Difference with Self is Empty Set |
By Real Number Line is Complete Metric Space and Baire Category Theorem, $\struct {\R, \tau}$ is a Baire space.
By definition of Baire Space, $\ds \bigcap_{i \mathop \in \N} U_i$ is dense in $\R$.
But Empty Set is Nowhere Dense.
This is a contradiction.
Hence $\Q$ is not a $G_\delta$ set in $\R$.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $30$. The Rational Numbers: $2$