Loop space of a topological space

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Definition

For a topological space X, the loop space is defined as the space of all continuous maps from the circle to X under the compact-open topology.

The term loop space is also used for loop space of a based topological space, which is defined in the context of a based topological space as all the basepoint-preserving maps from a circle (with chosen basepoint) to X under the compact-open topology. This latter loop space is a subspace of the loop space discussed in this article. For a path-connected space, the loop space as a based topological space intersects every path component of the overall loop space.