Dowker space

A Dowker space is a topological space that is T₄ but not countably paracompact.

Equivalences

If X is a normal T1 space (a T₄ space), then the following are equivalent:

• X is a Dowker space
• The product of X with the unit interval is not normal. C. H. Dowker 1951^[1]
• X is not 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 M.E. Rudin constructed one^[2] in 1971. Rudin's counterexample is a very large space (of cardinality $\aleph_\omega^\omega$) and is generally not well-behaved. Zoltán Balogh gave the first ZFC construction^[3] of a small (cardinality continuum) example, which was more well-behaved than Rudin's. Using PCF theory, M. Kojman and S. Shelah constructed^[4] a Dowker space of cardinality $\aleph_{\omega+1}$.

References

1. ^ C.H. Dowker, On countably paracompact spaces, Can. J. Math. 3 (1951) 219-224. Zbl. 0042.41007
2. ^ M.E. Rudin, A normal space X for which X × I is not normal, Fundam. Math. 73 (1971) 179-186. Zbl. 0224.54019
3. ^ Z. Balogh, "A small Dowker space in ZFC", Proc. Amer. Math. Soc. 124 (1996) 2555-2560. Zbl. 0876.54016
4. ^ M. Kojman, S. Shelah: "A ZFC Dowker space in $\aleph_{\omega+1}$: an application of PCF theory to topology", Proc. Amer. Math. Soc., 126(1998), 2459-2465.