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 ${\displaystyle \aleph _{\omega }^{\aleph _{0}}}$). 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 ${\displaystyle \aleph _{\omega +1}}$ that is also Dowker.^[4]