Irrational Number Space is not Weakly Sigma-Locally Compact

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 weakly $\sigma$-locally compact.

Proof

From Irrational Number Space is not Locally Compact Hausdorff Space, $\struct {\R \setminus \Q, \tau_d}$ is not weakly locally compact.

Hence the result from definition of weakly $\sigma$-locally compact.

$\blacksquare$

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: $8$