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 spaces

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

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

The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the subspace consisting of the whole space minus the point $(Ω,ω)$ can be shown to be non-normal. Therefore the Tychonoff plank is not completely normal. This shows that a subspace of a normal space need not be normal.