# Connectedness of finite topological spaces

## notes for combinatrorics seminar

Posted by Steven Clontz on March 18, 2024

Here’s some rough notes on what I’ll talk about at the South Alabama combinatorics seminar on March 21.

• Motivation:
• Modeling finite topological spaces
• Set + topology.
• Ex: X=5, basis={{0,1},{1},{2,3,4},{3,4}}
• Specialization preorder.
• Ex: 0 ≺ 1, 2 ≺ 3, 2 ≺ 4, 3 ≺ 4 ≺ 3
• Directed graph* (where x->y->z implies x->z)
• Connected topology = connected digraph
• Biconnected (π-Base P44)
• Connected space such that given two disjoint connected subspaces, one is a singleton.
• Equivalently, for any partition of the space into two connected subspaces, one is a singleton.
• Totally disconnected (π-Base P47)
• Every subspace with more than one point is disconnected
• Has a dispersion point (π-Base P47)
• Connected space with a dispersion point, a point whose removal results in a totally disconnected space
• Ex: 0 ≺ 1,2,3,4 has 0 as a dispersion point
• It’s also biconnected: actually any space with a dispersion point is biconnected
• Question: among finite spaces, are biconnected spaces just the spaces with a dispersion point?
• Not quite: 0, 0 ≺ 1, and 0 ≺ 1 ≺ 2 are biconnected without dispersion points
• So we need an extra requirement, one that we hadn’t ever thought to track in the π-Base…
• Thm: For finite spaces with at least 4 points, biconnected is equivalent to having a dispersion point.
• We cannot have x ≺ y ≺ z. Since {y,z} is connected, it follows {x,w} is disconnected for