---
tag: T0AEF3Z8C9
kind: concept
name: Frölicher space
msc: [58A40]
sources:
  - url: https://en.wikipedia.org/wiki/Fr%C3%B6licher_space
same_as:
  wikidata: Q5506747
  wikipedia: 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|F45VQ47GPV]] and are used for infinite-dimensional spaces such as spaces of smooth maps.
They are closely related to [[diffeological spaces|7JS0JFR89E]] and form a cartesian closed category.

## Explanations

### L1 human

One can learn a lot about a space by testing it from two sides: with curves going into it and with functions going out of it.
A Frölicher space fixes both families so that they match perfectly.
A curve passes the test if every allowed function sees it as smooth, and a function passes if it sees every allowed curve as smooth.
For ordinary Euclidean space, the usual smooth curves and smooth functions match in this way.