Tychonoff plank

Short description: Topological space in mathematics

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 $[0,{\omega }_1]$ 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 $\mathrm{\infty }=({\omega }_1,\omega )$.

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

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

See also

• List of topologies

References

• Kelley, John L. (1975), General Topology, Graduate Texts in Mathematics, 27 (1 ed.), New York: Springer-Verlag, Ch. 4 Ex. F, ISBN 978-0-387-90125-1 
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995), Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3 
• Willard, Stephen (1970), General Topology, Addison-Wesley, 17.12, ISBN 9780201087079, https://archive.org/details/generaltopology00will_0/page/17