Coarse Proximity, Proximity at Infinity, and Coarse Structures

In this dissertation, we introduce coarse proximities, explore some of their applications (e.g., proximity at infinity), and study the relationships between three different structures capturing large-scale properties of spaces: coarse proximities, asymptotic resemblances, and coarse space structures.After a short introduction to coarse topology and small-scale proximities, we recall basic definitions and theorems related to coarse spaces, asymptotic resemblance spaces, and bornologies. Then we investigate metric coarse proximities and introduce a general definition of coarse proximities. After exploring a few of their basic properties, we introduce coarse neighborhoods and use them to give an alternative definition of coarse proximities. We then proceed to show that coarse proximities induce weak asymptotic resemblances, and we use this fact to investigate coarse proximity maps to build a category of coarse proximity spaces whose morphisms are closeness classes of coarse proximity maps.Next we restrict our attention to the metric case and we construct a natural small-scale proximity structure on the set of unbounded subsets of a metric space. We also show how this structure naturally induces a small-scale proximity on the equivalence classes of the weak asymptotic resemblance induced by the metric. We call this space the “proximity space at infinity.” We then proceed to show that the construction is functorial, making up a functor from the category of unbounded metric spaces whose morphisms are closeness classes of coarse proximity maps (equivalently, coarse maps or asymptotic resemblance maps) to the category of proximity spaces whose morphisms are proximity maps.Finally, we investigate the relationships between coarse proximities, (weak) asymptotic resemblances, and coarse spaces structures. We also explore coarse and asymptotic normality and we show that under mild conditions, both normal coarse spaces and normal asymptotic resemblance spaces induce coarse proximities.