Rational Number Space is Completely Normal

Theorem

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.

Then $\struct {\Q, \tau_d}$ is a completely normal space.

Proof

From Euclidean Space is Complete Metric Space, a Euclidean space is a metric space.

From Metric Space fulfils all Separation Axioms it follows that $\struct {\Q, \tau_d}$ is a completely normal 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: $4$