Abstract by Matthew McGrath
A topological version of Hadwiger-Nelson
We call a surjective function, f, from a metric space (X,d) to a finite set, C, a "coloring" if d(x,y)=1 implies that f(x)≠f(y). If all colorings have |C|≥n then we say that n is the chromatic number of (X,d). The Hadwiger-Nelson problem asks what the chromatic number of the plane is with the euclidean metric. In order to explore topological versions of the Hadwiger-Nelson problem, we introduce the notion of a stable coloring and the stable chromatic number of a metric space. We will then prove that the limit of a sequence of metric spaces with the same stable chromatic number, n, is a metric space with a stable chromatic number less than or equal to n.