Set-theoretic topology

In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC).

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

Dowker conjectured that there were no Dowker spaces, and the conjecture was not resolved until M.E. Rudin constructed one in 1971. Rudin's counterexample is a very large space (of cardinality ${\displaystyle \aleph _{\omega }^{\aleph _{0}}}$) and is generally not well-behaved. 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 ${\displaystyle \aleph _{\omega +1}}$ that is also Dowker.

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