In mathematics, pointless topology (also called point-free or pointfree topology, or locale theory) is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann. The ideas of pointless topology are closely related to mereotopologies in which regions (sets) are treated as foundational without explicit reference to underlying point sets. == General concepts == Traditionally, a topological space consists of a set of points together with a topology, a system of subsets called open sets that with the operations of intersection and union for...

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). == Objects studied in set-theoretic topology == === Dowker spaces === 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 ...