  We rewrite classical topological definitions using the category-theoretic
notation of arrows and are led to concise reformulations in terms of simplicial
categories and orthogonality of morphisms, which we hope might be of use in the
formalisation of topology and in developing the tame topology of Grothendieck.
  Namely, we observe that topological and uniform spaces are simplicial objects
in the same category, a category of filters, and that a number of elementary
properties can be obtained by repeatedly passing to the left or right
orthogonal (in the sense of Quillen model categories) starting from a simple
class of morphisms, often a single typical (counter)example appearing
implicitly in the definition.
  Examples include the notions of: compact, discrete, connected, and totally
disconnected spaces, dense image, induced topology, and separation axioms, and,
outside of topology, finite groups being nilpotent, solvable, torsion-free,
p-groups, and prime-to-p groups; injective and projective modules; injective
and surjective (homo)morphisms.
