Loop Space

Let $Y^X$ be the set of continuous mappings $f:X\to Y$. Then the Topological Space for $Y^X$ supplied with a compact-open topology is called a Mapping Space, and if $Y=I$ is taken as the interval $(0,1)$, then $Y^I=\Omega(Y)$ 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.