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 $\aleph _{\omega }^{\aleph _{0}}$ ). 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. 73 (2). Polish Academy of Sciences: 179–186. doi:10.4064/fm-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. doi:10.1090/S0002-9939-96-03610-6. 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. 126 (8). American Mathematical Society: 2459–2465. doi:10.1090/S0002-9939-98-04884-9. Retrieved March 29, 2015.

[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. 73 (2). Polish Academy of Sciences: 179–186. doi:10.4064/fm-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. doi:10.1090/S0002-9939-96-03610-6. 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. 126 (8). American Mathematical Society: 2459–2465. doi:10.1090/S0002-9939-98-04884-9. Retrieved March 29, 2015.

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