Dowker space explained

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.

	\aleph₀
{\aleph
	\omega}

). 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

\aleph_\omega+1

that is also Dowker.^[4]

Notes and References

Dowker . C. H. . Clifford Hugh Dowker . 1951 . On countably paracompact spaces . . 3 . 219–224 . 10.4153/CJM-1951-026-2 . 0042.41007 . March 29, 2015 . July 14, 2014 . https://web.archive.org/web/20140714144135/http://cms.math.ca/cjm/v3/cjm1951v03.0219-0224.pdf . dead .
Rudin . Mary Ellen . Mary Ellen Rudin . 1971 . A normal space X for which X × I is not normal . . Polish Academy of Sciences . 73 . 2 . 179–186 . 10.4064/fm-73-2-179-186 . 0224.54019 . March 29, 2015.
Balogh . Zoltan T. . Zoltán Tibor Balogh . August 1996 . A small Dowker space in ZFC . . 124 . 8 . 2555–2560 . 10.1090/S0002-9939-96-03610-6 . 0876.54016 . March 29, 2015.
Kojman . Menachem . Shelah . Saharon . Saharon Shelah . 1998 . A ZFC Dowker space in
\aleph_\omega+1

: an application of PCF theory to topology . . American Mathematical Society . 126 . 8 . 2459–2465 . 10.1090/S0002-9939-98-04884-9 . March 29, 2015.