Localization of Large Scale Structures

We begin by giving the definition of coarse structures by John Roe, but quickly move to the equivalent concept of large scale geometry given by Jerzy Dydak. Next we present some basic but often used concepts and results in large scale geometry. We then state and prove the equivalence of various definitions of asymptotic dimension for arbitrary large scale spaces. Some of these are generalizations of asymptotic dimension for metric spaces, and many of the proofs are new. Particularly useful in proving the equivalences of the various definitions is the notion of partitions of unity, originally set forth by Jerzy Dydak. We then generalize the concept of bounded geometry, by defining the entropy and capacity of a set with respect to a cover. We show that all covers which are uniform with respect to a gauge form a large scale structure, which has many of the properties that spaces with bounded geometry have. Finally we restrict the uniformly bounded covers in a large scale structure in order to form a new structure called a localization. We seek to determine which large scale properties hold in the new structure.