Loop space

Remarks

Concatenation $(\alpha \cdot \beta)(s) = \alpha(2s)$ for $s \le 1/2$ and $\beta(2s - 1)$ for $s \ge 1/2$ is a continuous product on $\Omega X$ that is associative and unital up to homotopy. The set of path components $\pi_0(\Omega X)$ is in bijection with $\pi_1(X, x_0)$, and $\pi_n(\Omega X) \cong \pi_{n+1}(X, x_0)$ for $n \ge 1$. For pointed spaces $A$ and $X$, pointed homotopy classes of maps $\Sigma A \to X$ from the reduced suspension correspond bijectively to pointed homotopy classes of maps $A \to \Omega X$. Example: if $X$ is the one-point space, then $\Omega X$ is a single point, the constant loop. Non-example: the free loop space of all continuous maps $S^1 \to X$ with no condition at a base point is a different space; it contains loops that miss $x_0$.