# Unifying large scale and small scale geometry

## TL;DR

This paper introduces a unifying framework that connects small scale and large scale geometries through orthogonality relations, generalizing various compactification methods in topology.

## Contribution

It proposes a novel approach using orthogonality relations to unify different geometric and topological structures and compactifications.

## Key findings

- Unified framework for small and large scale geometries
- Generalized large scale compactifications including known types
- Connections between topology, proximity, and coarse spaces

## Abstract

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an orthogonality relation on subsets of $X$. The optimal case is if the orthogonality relation satisfies a variant of parallel-perpendicular decomposition from linear algebra. We show that this concept unifies small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). Using orthogonality relations we define large scale compactifications that generalize all well-known compactifications: Higson corona, Gromov boundary, \v{C}ech-Stone compactification, Samuel-Smirnov compactification, and Freudenthal compactification.