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 .

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

  • Kelley, John L. (1975), General Topology, Graduate Texts in Mathematics, 27 (1 ed.), New York: Springer-Verlag, Ch. 4 Ex. F, ISBN 978-0-387-90125-1, MR 0370454
  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
  • Willard, Stephen (1970), General Topology, Addison-Wesley, 17.12, ISBN 9780201087079, MR 0264581
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