# Dowker space

In the mathematical field of general topology, a Dowker space is a topological space that is T4 but not countably paracompact.

## Equivalences

C. H. Dowker showed, in 1951, the following:

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

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 $\aleph_\omega^{\aleph_0}$) 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 subspace of Rudin's Dowker space of cardinality $\aleph_{\omega+1}$ that is also Dowker.

## 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.