title: Dowker space
text: # Dowker space
  
  In the mathematical field of general topology, a Dowker space is a topological space that is $T_4$ 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 $T_1$ space (that is, a $T_4$ 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 $\aleph _{\omega }^{\aleph _{0}}$). 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 $\aleph _{\omega +1}$ that is also Dowker.
  
  