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.^[1] 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 ${\displaystyle \{\infty \}}$ is closed but not a G_δ set.

The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.^[2]

Notes

1. Steen & Seebach 1995, Example 86, item 2.
2. Walker, R. C. (1974). The Stone-Čech Compactification. Springer. pp. 95–97. ISBN 978-3-642-61935-9.

See also

• List of topologies

References

• Kelley, John L. (1975), General Topology, Graduate Texts in Mathematics, vol. 27 (1 ed.), New York: Springer-Verlag, Ch. 4 Ex. F, ISBN 978-0-387-90125-1, MR 0370454
• 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 0507446
• Willard, Stephen (1970), General Topology, Addison-Wesley, 17.12, ISBN 9780201087079, MR 0264581