Tychonoff plank
From Infogalactic: the planetary knowledge core
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 Gδ space: the singleton is closed but not a Gδ set.
References
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External links
- Barile, Margherita, "Tychonoff Plank", MathWorld.