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