Boundary of Rational Numbers

Let's consider the set of rational numbers $\mathbb{Q}$ within the standard topology on $\mathbb{R}$, which is the topology of the real line.

In this exercise, we need to determine the boundary of the rational numbers.

The boundary of $\mathbb{Q}$, denoted as $\partial \mathbb{Q}$, consists of points that are in both the closure of $\mathbb{Q}$ and the closure of the complement of $\mathbb{Q}$.

$$ \partial \mathbb{Q} = \text{Cl}(\mathbb{Q}) \cap \text{Cl}(\mathbb{R} - \mathbb{Q}) $$

The complement of the rational numbers $\mathbb{R} - \mathbb{Q}$ is the set of irrational numbers $\mathbb{I}$.

$$ \partial \mathbb{Q} = \text{Cl}(\mathbb{Q}) \cap \text{Cl}(\mathbb{I}) $$

The closure of the rational numbers is the set of real numbers $\text{Cl}(\mathbb{Q}) = \mathbb{R}$.

$$ \partial \mathbb{Q} = \mathbb{R} \cap \text{Cl}(\mathbb{I}) $$

The closure of $\mathbb{Q}$ is the set of all points in $\mathbb{R}$ that are limits of sequences of points in $\mathbb{Q}$. Since rational numbers are dense in the real line, it means that every real number can be approximated arbitrarily well by rational numbers. Therefore, the closure of $\mathbb{Q}$ is the entire real line: $$ \text{Cl}(\mathbb{Q}) = \mathbb{R} $$

For the same reason, the closure of the set of irrational numbers $\text{Cl}(\mathbb{I}) = \mathbb{R}$ is also the set of real numbers $\mathbb{R}$.

$$ \partial \mathbb{Q} = \mathbb{R} \cap \mathbb{R} $$

Therefore, the boundary of $\mathbb{Q}$ is the set of real numbers $\mathbb{R}$.

$$ \partial \mathbb{Q} = \mathbb{R} $$

In other words, every point on the real line is "on the boundary" between the set of rational numbers and its complement (the irrational numbers).

This means that every real number can be approximated by both rational and irrational numbers.

Alternative Solution

The boundary of a set is the difference between its closure and its interior.

$$ \partial \mathbb{Q} = \text{Cl}(\mathbb{Q}) - \text{Int}(\mathbb{Q}) $$

For the set of rational numbers, the closure is the set of all real numbers, $\text{Cl}(\mathbb{Q}) = \mathbb{R}$.

$$ \partial \mathbb{Q} = \mathbb{R} - \text{Int}(\mathbb{Q}) $$

The interior of the set of rational numbers is empty, $\text{Int}(\mathbb{Q}) = \emptyset$.

$$ \partial \mathbb{Q} = \mathbb{R} - \emptyset $$

The interior of $\mathbb{Q}$ consists of all points in $\mathbb{Q}$ that have a neighborhood fully contained within $\mathbb{Q}$. Since $\mathbb{Q}$ is scattered across the real line and every neighborhood of a rational number includes both rational and irrational numbers, no neighborhood is entirely contained within $\mathbb{Q}$. Therefore, $\text{Int}(\mathbb{Q}) = \emptyset$ (the empty set).

Thus, the boundary of the set of rational numbers is the set of all real numbers.

$$ \partial \mathbb{Q} = \mathbb{R} $$

And so forth.