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 [math]\displaystyle{ [0,\omega_1] }[/math] and [math]\displaystyle{ [0,\omega] }[/math], where [math]\displaystyle{ \omega }[/math] is the first infinite ordinal and [math]\displaystyle{ \omega_1 }[/math] the first uncountable ordinal. The deleted Tychonoff plank is obtained by deleting the point [math]\displaystyle{ \infty = (\omega_1,\omega) }[/math].
Properties
The 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 [math]\displaystyle{ \{\infty\} }[/math] is closed but not a Gδ set.
The Stone–Čech compactification of the deleted Tychonoff plank is the Tychonoff plank.[2]
Notes
- ↑ Steen & Seebach 1995, Example 86, item 2.
- ↑ Walker, R. C. (1974) (in en). The Stone-Čech Compactification. Springer. pp. 95–97. ISBN 978-3-642-61935-9. https://books.google.com/books?id=zhP2CAAAQBAJ&pg=PA95.
See also
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
- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995), Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3
- Willard, Stephen (1970), General Topology, Addison-Wesley, 17.12, ISBN 9780201087079, https://archive.org/details/generaltopology00will_0/page/17
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