Set-theoretic methods in Topology

Lecture Notes

Preliminaries

A few notions and results from Set Theory that will be used in the rest of the course: well-orderings, ordinals, cardinals, combinatorics, trees.

Elementarity

An introduction to the use of elementary substructures of the universe in set theory and topology.

Arkhangelskii's theorem

Arkhangelskii's theorem on the cardinality of first-countable spaces is a very good example of elementarity at work. We present Arkhangelskii's own proof, the Pol-Sapirovskii argument and the elementary version of the latter.

Dowker spaces

The rest of the notes will be devoted to Dowker spaces. This is a quick introduction to the subject.

Balogh's Dowker space

This Dowker space uses the machinery of elementary substructures in its construction.

Rudin's Dowker space

The first real Dowker space.

A Dowker space of size aleph_omega+1

A well-chosen subspace of Rudin's Dowker space.