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.
Definition
Let be the set of ordinals which are less than or equal to and the set of ordinals less than or equal to. The Tychonoff plank is defined as the set with the product topology.The deleted Tychonoff plank is the subset, where is the plank with a corner removed.
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 Gδ space: the singleton is closed but not a Gδ set.The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.