Rational Number Space is Sigma-Compact

Theorem

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

Then $\struct {\Q, \tau_d}$ is $\sigma$-compact.

Proof

From Rational Numbers are Countably Infinite, $\Q$ is countable.

Hence the result from definition of Countable Space is $\sigma$-Compact.

$\blacksquare$

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: $8$