# 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
and
, where
is the first infinite ordinal and
the first uncountable ordinal. The **deleted Tychonoff plank** is obtained by deleting the point
.

## 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
is closed but not a G_{δ} set.

## 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, 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

## External links

- Barile, Margherita. "Tychonoff Plank".
*MathWorld*.

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