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
- Set + topology.
- 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