Let be the set of continuous mappings . Then the Topological Space for supplied with a compact-open topology is called a Mapping Space, and if is taken as the interval , then is called a loop space (or Space of Closed Paths).
See also Machine, Mapping Space, May-Thomason Uniqueness Theorem
References
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.