Combinatorics of Finite Open Covers

For a Tychonoff space X, any finite cozero open cover U of X defines an abstract simplicial complex. We can then analyze various graph invariants of the 1-skeleton of this simplicial complex. By considering all finite open covers and their refinements, we can lift any graph invariant to define a corresponding topological graph invariant and determine its value for X. In this talk, we discuss specifically how to define these topological graph invariants. We also discuss several of these invariants in detail, including the topological chromatic number, clique number, chromatic index, maximum degree, independence number, and independence ratio. We will conclude with some discussion of ideas for future work.

(Note: This serves as my Overview presentation, so that may be relevant information to include as well).