Loop space

In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of based maps from the circle S to X with the compact-open topology. Two elements of a loop space can be naturally concatenated. With this concatenation operation, a loop space is an A_∞-space. The adjective A_∞ describes the manner in which concatenating loops is homotopy coherently associative.