Rational Number Space is Topological Space/Proof 1

Theorem

Let $\struct {\Q, \tau_d}$ be the rational number space formed by the rational numbers $\Q$ under the usual (Euclidean) topology $\tau_d$.

Then $\tau_d$ forms a topology.

Proof

From Rational Numbers form Metric Space we have that $\Q$ is a metric space under the Euclidean metric.

From Metric Induces Topology, it follows that the Euclidean topology forms a topology on $\Q$.

$\blacksquare$