Dowker space
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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.
- 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. Rudin's counterexample is a very large space (of cardinality ℵ ω ℵ 0 {\displaystyle \aleph _{\omega }^{\aleph _{0}}}). Zoltán Balogh gave the first ZFC construction of a small (cardinality continuum) example, 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 ℵ ω + 1 {\displaystyle \aleph _{\omega +1}} that is also Dowker.