Dowker space
In the mathematical field of general topology, a Dowker space is a topological space that is T4 but not countably paracompact. They are named after Clifford Hugh Dowker. The non-trivial task of providing an example of a Dowker space (and therefore also proving their existence as mathematical objects) helped mathematicians better understand the nature and variety of topological spaces.
Equivalences
Dowker showed, in 1951, the following:
If X is a normal T1 space (that is, a T4 space), then the following are equivalent:
- X is a Dowker space
- The product of X with the unit interval is not normal.[1]
- X is not countably metacompact.
Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until Mary Ellen Rudin constructed one in 1971.[2] Rudin's counterexample is a very large space (of cardinality [math]\displaystyle{ \aleph_\omega^{\aleph_0} }[/math]). Zoltán Balogh gave the first ZFC construction of a small (cardinality continuum) example,[3] which was more well-behaved than Rudin's. Using PCF theory, M. Kojman and S. Shelah constructed a subspace of Rudin's Dowker space of cardinality [math]\displaystyle{ \aleph_{\omega+1} }[/math] that is also Dowker.[4]
References
- ↑ Dowker, C. H. (1951). "On countably paracompact spaces". Can. J. Math. 3: 219–224. doi:10.4153/CJM-1951-026-2. http://cms.math.ca/cjm/v3/cjm1951v03.0219-0224.pdf. Retrieved March 29, 2015.
- ↑ Rudin, Mary Ellen (1971). "A normal space X for which X × I is not normal". Fundam. Math. (Polish Academy of Sciences) 73 (2): 179–186. doi:10.4064/fm-73-2-179-186. http://matwbn.icm.edu.pl/ksiazki/fm/fm73/fm73121.pdf. Retrieved March 29, 2015.
- ↑ Balogh, Zoltan T. (August 1996). "A small Dowker space in ZFC". Proc. Amer. Math. Soc. 124 (8): 2555–2560. doi:10.1090/S0002-9939-96-03610-6. https://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03610-6/S0002-9939-96-03610-6.pdf. Retrieved March 29, 2015.
- ↑ Kojman, Menachem; Shelah, Saharon (1998). "A ZFC Dowker space in [math]\displaystyle{ \aleph_{\omega+1} }[/math]: an application of PCF theory to topology". Proc. Amer. Math. Soc. (American Mathematical Society) 126 (8): 2459–2465. doi:10.1090/S0002-9939-98-04884-9. https://www.ams.org/proc/1998-126-08/S0002-9939-98-04884-9/S0002-9939-98-04884-9.pdf. Retrieved March 29, 2015.
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