Rational Number Space is not Scattered

Theorem

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

Then $\struct {\Q, \tau_d}$ is not scattered.

Proof

For a space to be scattered, it needs by definition to have no subset which is dense-in-itself.

From Rational Number Space is Dense-in-itself, $\struct {\Q, \tau_d}$ is dense-in-itself.

Hence the result.

$\blacksquare$

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: $9$