BGU Seminar in Logic, Set Theory and Topology
Title: Elementary topology via finite topological spaces
We observe that several elementary definitions in point-set topology
can be reformulated in terms of finite topological spaces
and elementary category theory. This includes compactness
of Hausdorff spaces, being connected, discrete, the separation axioms.
Though elementary, these observations raise a few open questions.
For example, I was not able to prove that this reformulation of
compactness gives the correct answer for non-Hausdorff spaces,
or whether implications between various topological properties
can also be proved entirely in terms of finite topological spaces,
without any additional axioms.