In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Frölicher spaces extend the notions of calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

 and smooth manifolds. They were introduced in 1982 by the mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 Alfred Frölicher.

Definition

A Frölicher space consists of a non-empty set X together with a subset C of Hom(R, X) called the set of smooth curves, and a subset F of Hom(X, R) called the set of smooth real functions, such that for each real function

f : X → R

in F and each curve

c : R → X

in C, the following axioms are satisfied:

1. f in F if and only if for each γ in C, f . γ in C^∞(R, R)
2. c in C if and only if for each φ in F, φ . c in C^∞(R, R)

Let A and B be two Frölicher spaces. A map

m : A → B

is called smooth if for each smooth curve c in C_A, m.c is in C_B. Furthermore the space of all such smooth maps has itself the structure of a Frölicher space. The smooth functions on

C^∞(A, B)

are the images of