Path 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. If (Y,*) is a pointed space, then the mapping space (Y,*)^((I,0))=PY of pointed maps is called the path space of Y. In words, PY is the space of all paths which begin at *. PY is a contractible space with the contraction H:PY×[0,1]->PY given by H(lambda,t)=lambda(1-t).

See also

Loop Space, Mapping Space

This entry contributed by John Renze

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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.

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Path Space

Cite this as:

Renze, John. "Path Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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