Frölicher space


In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. They were introduced in 1982 by the mathematician Alfred Frölicher.

Definition

A Frölicher space consists of a non-empty set X together with a subset C of Hom called the set of smooth curves, and a subset F of Hom called the set of smooth real functions, such that for each real function
in F and each curve
in C, the following axioms are satisfied:
  1. f in F if and only if for each γ in C, in C∞
  2. c in C if and only if for each φ in F, in C∞
Let A and B be two Frölicher spaces. A map
is called smooth if for each smooth curve c in CA, is in CB. Furthermore, the space of all such smooth maps has itself the structure of a Frölicher space. The smooth functions on
are the images of