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 ${\displaystyle \infty =(\omega ,\omega _{1})}$.

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 G_δ space: the singleton $\{\infty \}$ is closed but not a G_δ set.

References

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

External links

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