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  "body": "\n## Definition\n\nLet $G$ be a topological group.\nA *classifying space* of $G$ is the base space $B$ of a [[universal principal $G$-bundle|1HDH716Z24]] $E \\to B$.\nIt is written $BG$, and the total space is written $EG$.\n\n## Remarks\n\nAny two classifying spaces of $G$ are homotopy equivalent, so $BG$ is well defined up to homotopy.\nFor a CW complex $X$, isomorphism classes of [[principal $G$-bundles|8E37E74XVP]] over $X$ correspond to homotopy classes of maps $X \\to BG$.\nA principal $G$-bundle over a CW complex with [[contractible|R5WP8Z69Q8]] total space is universal, so its base is a $BG$.\n\nExample: the circle group $S^1$ acts freely on the unit sphere $S^\\infty$ of $\\C^\\infty$, with quotient $\\C P^\\infty$.\nSince $S^\\infty$ is contractible, $\\C P^\\infty$ is a classifying space of $S^1$.\n\nNon-example: $\\C P^1$ is not a classifying space of $S^1$.\nThe bundle $S^3 \\to \\C P^1$ is a principal $S^1$-bundle, but its total space $S^3$ is not contractible, and $\\pi_3(\\C P^1) \\cong \\Z$ while $\\pi_3(\\C P^\\infty) = 0$.\n\nFor a discrete group $G$, a classifying space is the same as a [[classifying space of the group|WA796ZCNRX]], an Eilenberg–MacLane space $K(G, 1)$.\nOther examples: $BO(n)$ and $BU(n)$ are infinite Grassmannians, and $B\\Z = S^1$.\n",
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