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 $[0,\omega]$ and $[0,\omega_1]$, where $\omega$ is the first infinite ordinal and $\omega_1$ the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point $\infty = (\omega,\omega_1)$.

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.

External links

• Barile, Margherita, "Tychonoff Plank", MathWorld.