  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.
