Let  be the set of continuous mappings . Then the topological space  supplied with the compact-open topology is called a mapping space, and if  is taken as the circle , then  is called the "free loop space of " (or the space of closed paths).

If  is a pointed space, then a basepoint can be picked on the circle and the mapping space  of pointed maps can be formed. This space is denoted  and is called the "loop space of ."

REFERENCES:

Bredon, G. Topology and Geometry New York: Springer-Verlag, p. 456, 1993.

Brylinski, J.-L. Loop Spaces, Characteristic Classes and Geometric Quantization. Boston, MA: Birkhäuser, 1993.

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 658, 1980.