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.{{cite journal |last=Dowker |first=C. H. |authorlink=Clifford Hugh Dowker |date=1951 |title=On countably paracompact spaces |journal=Can. J. Math. |volume=3 |pages=219–224 |doi=10.4153/CJM-1951-026-2 |zbl=0042.41007 |url=http://cms.math.ca/cjm/v3/cjm1951v03.0219-0224.pdf |accessdate=March 29, 2015 |archive-date=July 14, 2014 |archive-url=https://web.archive.org/web/20140714144135/http://cms.math.ca/cjm/v3/cjm1951v03.0219-0224.pdf |url-status=dead }}
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.{{cite journal |last=Rudin |first=Mary Ellen |authorlink=Mary Ellen Rudin |date=1971 |title=A normal space X for which X × I is not normal |journal=Fundam. Math. |publisher=Polish Academy of Sciences |volume=73 |number=2 |pages=179–186 |doi=10.4064/fm-73-2-179-186 |zbl=0224.54019 |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm73/fm73121.pdf |accessdate=March 29, 2015}} Rudin's counterexample is a very large space (of cardinality $\aleph_\omega^{\aleph_0}$ ). Zoltán Balogh gave the first ZFC construction of a small (cardinality continuum) example,{{cite journal |last=Balogh |first=Zoltan T. |authorlink=Zoltán Tibor Balogh |date=August 1996 |title=A small Dowker space in ZFC |journal=Proc. Amer. Math. Soc. |volume=124 |number=8 |pages=2555–2560 |doi=10.1090/S0002-9939-96-03610-6 |zbl=0876.54016 |url=https://www.ams.org/journals/proc/1996-124-08/S0002-9939-96-03610-6/S0002-9939-96-03610-6.pdf |accessdate=March 29, 2015}} 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 $\aleph_{\omega+1}$ that is also Dowker.{{cite journal |last1=Kojman |first1=Menachem |last2=Shelah |first2=Saharon |authorlink2=Saharon Shelah |date=1998 |title=A ZFC Dowker space in $\aleph_{\omega+1}$ : an application of PCF theory to topology |journal=Proc. Amer. Math. Soc. |publisher=American Mathematical Society |volume=126 |number=8 |pages=2459–2465 |doi=10.1090/S0002-9939-98-04884-9 |url=https://www.ams.org/proc/1998-126-08/S0002-9939-98-04884-9/S0002-9939-98-04884-9.pdf |accessdate=March 29, 2015}}

References

Category:Properties of topological spaces

Category:Separation axioms