Musings on general topology and orderings

The motivation for this post is that these days I am goofing around with lattices arising from subspaces of hypergraphs, which are complete and atomistic.

Motivation.

Given a set and , we let and .

Motivation.

Let denote the collection of infinite subsets of (the first infinite ordinal) and let be any map.

Let denote the first infinite cardinal – that is, the set of non-negative integers.

Suppose you want to have a graph with chromatic number equaling some value , such that is minimal with this property.

The starting point of this blog post is a slight reformulation of the separation axiom: A topological space is if for all there is a set such that Given a cardinal , we say that a space is if for all subsets with there is a set such that “splits” , or more

For we write if is finite, and we write if and .

Many people I talk to about graph theory feel some uneasiness when it comes to the notion of “minor”.