Infinite loop space

Remarks

The sequence $(X_n)$ with the adjoint maps $\Sigma X_n \to X_{n+1}$ is an $\Omega$-spectrum $E$, and $X$ is weakly equivalent to its zeroth space $\Omega^\infty E$. Conversely, the zeroth space of every $\Omega$-spectrum is an infinite loop space.

Example: for an abelian group $A$, the Eilenberg–MacLane spaces satisfy $K(A, n) \simeq \Omega K(A, n+1)$, so $K(A, 0) = A$ with the discrete topology is an infinite loop space with $X_n = K(A, n)$. The check: $\pi_k \Omega K(A, n+1) = \pi_{k+1} K(A, n+1)$, which is $A$ for $k = n$ and $0$ otherwise.

Example: for a pointed CW complex $Y$, the space $QY = \operatorname{colim}_n \Omega^n \Sigma^n Y$ is an infinite loop space, with $X_n = Q \Sigma^n Y$. By Bott periodicity, $\Z \times BU$ is an infinite loop space too.

Non-example: the sphere $S^2$ is not an infinite loop space, and not even a loop space. A loop space is an H-space, and $S^2$ is not one: a cup product computation in $H^*(S^2 \times S^2; \Z)$ rules out any H-space multiplication on $S^2$.