TOPICS
Search

Loop Space


Let Y^X be the set of continuous mappings f:X->Y. Then the topological space Y^X supplied with the compact-open topology is called a mapping space, and if X=I is taken as the circle S^1, then Y^I=LY is called the "free loop space of Y" (or the space of closed paths).

If (Y,*) is a pointed space, then a basepoint can be picked on the circle and the mapping space (Y,*)^((S^1,*)) of pointed maps can be formed. This space is denoted OmegaY and is called the "loop space of Y."


See also

Machine, Mapping Space, May-Thomason Uniqueness Theorem, Path Space, Pointed Space

This entry contributed by John Renze

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Loop Space

Cite this as:

Renze, John. "Loop Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LoopSpace.html

Subject classifications