en>John of Reading: Typo/general fixing, replaced: of the of the → of the using AWB

2013-11-18T17:11:06Z

Typo/general fixing, replaced: of the of the → of the using AWB

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{{Context|date=August 2009}}
In the [[mathematics|mathematical]] field of [[general topology]], a '''Dowker space''' is a [[topological space]] that is [[normal space|T<sub>4]]</sub> but not [[paracompact space|countably paracompact]].

==Equivalences==

If ''X'' is a normal [[T1 space]] (a T<sub>4</sub> space), then the following are equivalent:
* ''X'' is a Dowker space
* The product of ''X'' with the unit interval is not normal. [[Clifford Hugh Dowker|C. H. Dowker]] 1951<ref>
C.H. Dowker, On countably paracompact spaces, ''[[Canadian Journal of Mathematics|Can. J. Math.]]'' '''3''' (1951) 219-224. [[Zentralblatt MATH|Zbl.]] 0042.41007</ref>
* ''X'' is not [[Metacompact space|countably metacompact]]. This was also shown by Dowker, according to Balogh.

Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until [[Mary Ellen Rudin|M.E. Rudin]] constructed one<ref>M.E. Rudin, A normal space ''X'' for which ''X × I'' is not normal, ''[[Fundamenta Mathematicae|Fundam. Math.]]'' '''73''' (1971) 179-186. Zbl. 0224.54019</ref> in 1971. Rudin's counterexample is a very large space (of [[cardinality]] <math>\aleph_\omega^{\aleph_0}</math>) and is generally not [[well-behaved]]. [[Zoltán Tibor Balogh|Zoltán Balogh]] gave the first [[ZFC]] construction<ref>Z. Balogh, [http://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03610-6/S0002-9939-96-03610-6.pdf "A small Dowker space in ZFC"], ''[[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]]'' '''124''' (1996) 2555-2560. Zbl. 0876.54016</ref> of a small (cardinality [[Cardinality of the continuum|continuum]]) example, which was more [[well-behaved]] than Rudin's. Using [[PCF theory]], M. Kojman and [[Saharon Shelah|S. Shelah]] constructed<ref>M. Kojman, S. Shelah: [http://www.ams.org/proc/1998-126-08/S0002-9939-98-04884-9/S0002-9939-98-04884-9.pdf "A ZFC Dowker space in <math>\aleph_{\omega+1}</math>: an application of PCF theory to topology"], ''Proc. Amer. Math. Soc.'', '''126'''(1998), 2459-2465.</ref> a subspace of Rudin's Dowker space of cardinality <math>\aleph_{\omega+1}</math> that is also Dowker.

==References==
<references/>

[[Category:Properties of topological spaces]]
[[Category:Separation axioms]]

Small-bias sample space - Revision history

en>John of Reading: Typo/general fixing, replaced: of the of the → of the using AWB