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 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.

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.