If $X=[0,1]^{[0,1]}$ in product topology, then it is a compact Hausdorff space which is not second countable. However I think this space is infinite in (covering, inductive)dimension. Does there exist a compact Hausdorff space which is not second countable and finite in dimension?
2026-05-06 05:10:40.1778044240
finite dimensional compact Hausdorff space, not second countable
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$X=\omega_1 + 1$ is not second countable, compact and hereditarily normal, and has $\dim(X)=\text{Ind}(X)=0$. I think $0$ counts as finite...
$\{0,1\}^{[0,1]}$ also has the same dimension values and is compact and non-second countable.