Loop space of a topological space

Definition
For a topological space $$X$$, the loop space is defined as the space of all defining ingredient::continuous maps from the defining ingredient::circle to $$X$$ under the defining ingredient::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.