Tychonoff plank explained

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 inf ty = (\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

\{ inf ty\} is closed but not a G_δ set.

The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.^[1]

See also

• List of topologies

Notes and References

1. Book: Walker, R. C.. The Stone-Čech Compactification. Springer. 1974. 978-3-642-61935-9. 95–97. en.