Coarse Proximity and Proximity at Infinity
University of Tennessee, Knoxville, TN, USA
Coarse topology (i.e., large scale geometry) is a branch of mathematics investigating large-scale properties of spaces. While classical topology is primarily concerned with what happens "on the small-scale" (e.g., limits, continuity), coarse topology focus predominantly on what happens "on the large-scale" (e.g., asymptotic dimension, coarse equivalence of spaces). The idea of “translating” a small-scale world to its large-scale counterpart has been extensively explored by coarse topologists. In this talk, we will focus on “coarsening” the notion of proximity. We will start with reviewing the notion of a proximity space and introducing the definition of a metric coarse proximity. After investigating a few properties of this relation, we will generalize the metric case to obtain coarse proximities on any set with bornology. Then we will proceed to show the existence of the category of coarse proximity spaces whose morphisms are closeness classes of coarse proximity maps. We will conclude with the construction of a “proximity space at infinity”- a coarse invariant of unbounded metric spaces and a link between the small-scale and the large-scale worlds. No prior knowledge of coarse topology will be assumed. Familiarity with metric spaces is desirable. This is joint work with Jeremy Siegert.