Rational Numbers form Metric Subspace of Real Numbers under Euclidean Metric
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Theorem
Let $\left({\Q, d_\Q}\right)$ be the set of rational numbers under the function $d_\Q: \Q \times \Q \to \R$ defined as:
- $\forall x, y \in \Q: d \left({x, y}\right) = \left\vert{x - y}\right\vert$
Then $\left({\Q, d_\Q}\right)$ is a metric subspace of $\left({\R, d}\right)$, where:
- $\R$ is the set of real numbers
- $d: \R \times \R \to \R$ is the Euclidean metric on $\R$.
Proof
From Rational Numbers form Subfield of Real Numbers:
- $\Q \subseteq \R$
By the definition of the Euclidean metric on $\R$:
- $\forall x, y \in \R: d \left({x, y}\right) = \left\vert{x - y}\right\vert$
and so $d_\Q$ is a restriction of $d$:
- $d_\Q = d {\restriction}_{\Q \times \Q}$
From Euclidean Metric on Real Number Line is Metric, $d$ is a metric on $\R$.
The result follows by definition of metric subspace.
$\blacksquare$
Sources
- 1962: Bert Mendelson: Introduction to Topology ... (previous) ... (next): $\S 2.7$: Subspaces and Equivalence of Metric Spaces: Example $1$