Tychonoff

Tychonoff plank

In topology, the Tychonoff plank is a topological space that is a counterexample to several plausible-sounding conjectures. It is defined as the product of the two ordinal spaces

$[0,\Omega]\times[0,\omega]$

where $ω$ is the first infinite ordinal and $Ω$ the first uncountable ordinal.

The deleted Tychonoff plank is obtained by deleting the point $\infty = (\Omega,\omega)$ .

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

References

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

External links

MathWorld on the Tychonoff plank

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