 Loop space

The loop space $\Omega X$ is the space of based loops in $X$. It is the homotopy fiber of the inclusion of the base point into $X$.

In mathematics, particularly in algebraic topology, the term "loop space" refers to a certain kind of space that captures the idea of loops in a given topological space. Specifically, the loop space of a pointed topological space \( (X, x_0) \) is the space of all loops based at the point \( x_0 \).