Tychonoff plank

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In topology, the Tychonoff plank is a topological space that is a counterexample to several plausible-sounding conjectures. It is defined as the product of the two ordinal space

$[0,\omega_1]\times[0,\omega]$

where $ω$ is the first infinite ordinal and $ω 1$ the first uncountable ordinal.

The deleted Tychonoff plank is obtained by deleting the point $\infty = (\Omega,\omega)$ .

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 Fσ set.