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 .
![{\displaystyle [0,\omega _{1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd08431e2f94a42dda5225e2cbb63f4712cdad30)
![{\displaystyle [0,\omega ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae067f10ed5571f292209c44c33592b19f2e5ea)
Properties
editThe Tychonoff plank is a compact Hausdorff space and is therefore a normal space. However, the deleted Tychonoff plank is non-normal.[1] 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.[2]
Notes
edit- ^ Steen & Seebach 1995, Example 86, item 2.
- ^ Walker, R. C. (1974). The Stone-Čech Compactification. Springer. pp. 95–97. ISBN 978-3-642-61935-9.
See also
editReferences
edit- Kelley, John L. (1975), General Topology, Graduate Texts in Mathematics, vol. 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