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  "body": "\n## Definition\n\nLet $X$ be a [[pointed space|T9DG8DDE75]], and write $\\Omega Y$ for the [[loop space|2JFF5F3EGF]] of a pointed space $Y$.\nThe space $X$ is an *infinite loop space* if there are pointed spaces $X_0, X_1, X_2, \\ldots$ and [[weak homotopy equivalences|4Q3WY1RX91]]\n$$X \\simeq X_0, \\qquad X_n \\to \\Omega X_{n+1} \\quad \\text{for all } n \\ge 0.$$\n\n## Remarks\n\nThe sequence $(X_n)$ with the adjoint maps $\\Sigma X_n \\to X_{n+1}$ is an $\\Omega$-[[spectrum|1GHF4EKBXS]] $E$, and $X$ is weakly equivalent to its zeroth space $\\Omega^\\infty E$.\nConversely, the zeroth space of every $\\Omega$-spectrum is an infinite loop space.\n\nExample: 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 $X_n = K(A, n)$.\nThe check: $\\pi_k \\Omega K(A, n+1) = \\pi_{k+1} K(A, n+1)$, which is $A$ for $k = n$ and $0$ otherwise.\n\nExample: for a pointed CW complex $Y$, the space $QY = \\operatorname{colim}_n \\Omega^n \\Sigma^n Y$ is an infinite loop space, with $X_n = Q \\Sigma^n Y$.\nBy Bott periodicity, $\\Z \\times BU$ is an infinite loop space too.\n\nNon-example: the sphere $S^2$ is not an infinite loop space, and not even a loop space.\nA loop space is an [[H-space|EASHTX8EG5]], 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 $S^2$.\n",
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