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 .

PropertiesEdit

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.[1]

NotesEdit

  1. ^ Walker, R. C. (1974). The Stone-Čech Compactification. Springer. pp. 95–97. ISBN 978-3-642-61935-9.

ReferencesEdit

External linksEdit