Frölicher space
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In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. They were introduced in 1982 by the mathematician Alfred Frölicher.
[edit] 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
and each curve
- c : R → X
- f in F if and only if for each γ in C, f . γ in C∞(R, R)
- 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 CA, m.c is in CB. 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
