Frölicher space

Summary

A Frölicher space is a set with a family of smooth curves into it and a family of smooth real functions on it. Each of the two families determines the other. A curve is smooth when every smooth function composed with it is smooth. A function is smooth when it composes smoothly with every smooth curve. Frölicher spaces generalize smooth manifolds and are used for infinite-dimensional spaces such as spaces of smooth maps. They are closely related to diffeological spaces and form a cartesian closed category.