Irrational Number Space is not Scattered
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Theorem
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Then $\struct {\R \setminus \Q, \tau_d}$ is not scattered.
Proof
For a space to be scattered, it needs by definition to have no subset which is dense-in-itself.
From Irrational Number Space is Dense-in-itself, $\struct {\R \setminus \Q, \tau_d}$ 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: $31$. The Irrational Numbers: $9$