# 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 ${\displaystyle [0,\omega _{1}]}$ and ${\displaystyle [0,\omega ]}$, where ${\displaystyle \omega }$ is the first infinite ordinal and ${\displaystyle \omega _{1}}$ the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point ${\displaystyle \infty =(\omega _{1},\omega )}$.

## 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 ${\displaystyle \{\infty \}}$  is closed but not a Gδ set.

The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.[1]

## Notes

1. ^ Walker, R. C. (1974). The Stone-Čech Compactification. Springer. pp. 95–97. ISBN 978-3-642-61935-9.