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

Definition

Let ${\displaystyle \Omega }$ be the set of ordinals which are less than or equal to ${\displaystyle \omega }$ and ${\displaystyle \Omega _{1}}$ the set of ordinals less than or equal to ${\displaystyle \omega _{1}}$. The Tychonoff plank is defined as the set ${\displaystyle \Omega \times \Omega _{1}}$ with the product topology. ^[2]

The deleted Tychonoff plank is the subset ${\displaystyle S=\Omega \times \Omega _{1}\setminus \{(\omega ,\omega _{1})\}}$, where ${\displaystyle S}$ is the plank with a corner removed. ^[3]

Properties

The Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal. ^[4] 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. ^[5]

The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank. ^[6]

See also

• List of topologies

References

1. Weisstein, Eric W. "Tychonoff Plank". MathWorld . Retrieved 20 July 2025.
2. 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.
3. 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.
4. Steen & Seebach 1995, Example 86, item 2.
5. Willard, Stephen (1970). General Topology . Addison-Wesley. 17.12. ISBN   9780201087079. MR   0264581.
6. Walker, R. C. (1974). The Stone-Čech Compactification. Springer. pp. 95–97. ISBN   978-3-642-61935-9.