Rational Numbers form Metric Space
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Theorem
Let $\Q$ be the set of all rational numbers.
Let $d: \Q \times \Q \to \R$ be defined as:
- $\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the absolute value of $x$.
Then $d$ is a metric on $\Q$ and so $\struct {\Q, d}$ is a metric space.
Proof
From the definition of absolute value:
- $\size {x_1 - x_2} = \sqrt {\paren {x_1 - x_2}^2}$
This is the same as the Euclidean metric.
This is shown in Euclidean Metric on Real Vector Space is Metric to be a metric.
From Rational Numbers form Vector Space, it follows that the set of all rational numbers is a 1-dimensional Euclidean space.
$\blacksquare$
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: $4$