Tychonoff plank

In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures. It is defined as the topological product of the two ordinal spaces $$[0,\omega_1]$$ and $$[0,\omega]$$, where $$\omega$$ is the first infinite ordinal and $$\omega_1$$ the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point $$\infty = (\omega_1,\omega)$$.

Properties
The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal. Therefore the Tychonoff plank is not completely normal. This shows that a subspace of a normal space need not be normal. The Tychonoff plank is not perfectly normal because it is not a Gδ space: the singleton $$\{\infty\}$$ is closed but not a Gδ set.

The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.