# Loop space of a topological space

From Topospaces

## Definition

For a topological space , the **loop space** is defined as the space of all continuous maps from the circle to 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 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.