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. Topological spaces are sets together with some subsets (designated as "open sets") satisfying certain properties. Topological spaces arose as generalization of the open sets of spaces studied in elementary mathematics, such as open disks in the Euclidean plane, open balls in the Euclidean space, and open intervals of the real line.

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 $\aleph _{\omega }^{\aleph _{0}}$ ). 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]

References

Dowker, C. H. (1951). "On countably paracompact spaces" (PDF). Can. J. Math. 3: 219–224. doi:10.4153/CJM-1951-026-2. Zbl 0042.41007. Retrieved March 29, 2015.
Rudin, Mary Ellen (1971). "A normal space X for which X × I is not normal" (PDF). Fundam. Math. Polish Academy of Sciences. 73 (2): 179–186. Zbl 0224.54019. Retrieved March 29, 2015.
Balogh, Zoltan T. (August 1996). "A small Dowker space in ZFC" (PDF). Proc. Amer. Math. Soc. 124 (8): 2555–2560. Zbl 0876.54016. Retrieved March 29, 2015.
Kojman, Menachem; Shelah, Saharon (1998). "A ZFC Dowker space in $\aleph _{\omega +1}$ : an application of PCF theory to topology" (PDF). Proc. Amer. Math. Soc. American Mathematical Society. 126 (8): 2459–2465. Retrieved March 29, 2015.

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[1] Dowker, C. H. (1951). "On countably paracompact spaces" (PDF). Can. J. Math. 3: 219–224. doi:10.4153/CJM-1951-026-2. Zbl 0042.41007. Retrieved March 29, 2015.

[2] Rudin, Mary Ellen (1971). "A normal space X for which X × I is not normal" (PDF). Fundam. Math. Polish Academy of Sciences. 73 (2): 179–186. Zbl 0224.54019. Retrieved March 29, 2015.

[3] Balogh, Zoltan T. (August 1996). "A small Dowker space in ZFC" (PDF). Proc. Amer. Math. Soc. 124 (8): 2555–2560. Zbl 0876.54016. Retrieved March 29, 2015.

[4] Kojman, Menachem; Shelah, Saharon (1998). "A ZFC Dowker space in $\aleph _{\omega +1}$ : an application of PCF theory to topology" (PDF). Proc. Amer. Math. Soc. American Mathematical Society. 126 (8): 2459–2465. Retrieved March 29, 2015.