General topology

Finite topological spaces are topological spaces with finitely many points. While this may sound like a useless notion, finite spaces connect topology with combinatorics, logic, and other parts of discrete mathematics. Via the Alexandroff topology , we have the dictionary:

Topology	Order theory
Finite spaces	Finite preorders
\(T_0\) finite spaces	Finite partial orders
Alexandroff spaces	Preorders
\(T_0\) Alexandroff spaces	Partial orders

In all cases, continuous functions are monotone maps. This correspondence and much more is described in:

• May, 2019, draft: Finite spaces and larger contexts (pdf)
  • Book draft prepared for May’s REU at U. Chicago, among other notes
  • May, 2010, notes: Finite topological spaces (pdf)
  • May, 2010, notes: Finite groups and finite spaces (pdf)
• Barmak, 2011: Algebraic topology of finite topological spaces and applications (doi)
• Speer, 2007: A short study of Alexandroff spaces (arxiv)
• Kong, Kopperman, Meyer, 1991: A topological approach to digital topology (doi)

The Alexandroff topology also shows up in books on locale theory, such as Johnstone’s and Vickers’ (see below).

Point-free topology , also known as pointless topology and locale theory, is a lattice-theoretic approach to topology not based on the idea of a topological space as a set of points equipped with extra structure.

• Johnstone, 1983: The point of pointless topology (doi)
• Johnstone, 1982: Stone spaces (nLab )
• Vickers, 1989: Topology via logic
  • In his review (doi), Johnstone describes Vickers’ book “as a predigested version of his own book Stone spaces,” which seems rather unfair. In a theme totally absent from Jonhstone’s book, Vickers applies topology to the denotational semantics of programming languages.
  • Vickers, 1999: Topology via constructive logic (pdf)
• Vickers, 2022: Generalized point-free spaces, pointwise (arxiv)
  • Survey of “Grothendieck’s generalized spaces” with a helpful glossary