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  "body": "\n## Definition\n\nLet $G$ be a topological group and let $p \\colon E \\to B$ be a [[principal $G$-bundle|8E37E74XVP]].\nFor a space $X$, let $\\operatorname{Prin}_G(X)$ be the set of isomorphism classes of principal $G$-bundles over $X$.\nLet $[X, B]$ be the set of [[homotopy|9YA6S6HD9V]] classes of continuous maps $X \\to B$.\nThe bundle $p$ is *universal* if, for every [[CW complex|5W95QFHXPR]] $X$, the map\n$$[X, B] \\to \\operatorname{Prin}_G(X), \\quad [f] \\mapsto [f^* E],$$\nis a bijection, where $f^* E$ is the pullback bundle.\n\n## Remarks\n\nThe map is well defined because pullbacks along homotopic maps are isomorphic over CW complexes.\nA universal bundle is written $EG \\to BG$, and $BG$ is a [[classifying space|WSZ40H5E4T]] of $G$; it is unique up to homotopy equivalence.\nA principal $G$-bundle over a CW complex whose total space is [[contractible|R5WP8Z69Q8]] is universal.\nSuch bundles exist for every topological group $G$, for instance by Milnor's join construction.\n\nExample: $\\Z/2\\Z$ acts freely on the infinite sphere $S^\\infty$ by $x \\mapsto -x$, and $S^\\infty \\to \\R P^\\infty$ is a principal $\\Z/2\\Z$-bundle.\nSince $S^\\infty$ is contractible, this bundle is universal.\n\nNon-example: the bundle $S^1 \\to \\R P^1$, $x \\mapsto \\set{x, -x}$, is a principal $\\Z/2\\Z$-bundle but not universal.\nTake $X = \\R P^2$: there are two principal $\\Z/2\\Z$-bundles over $\\R P^2$, the trivial one and the double cover $S^2 \\to \\R P^2$.\nBut every map $\\R P^2 \\to \\R P^1 \\cong S^1$ is null-homotopic, because $H^1(\\R P^2; \\Z) = 0$, so $[\\R P^2, \\R P^1]$ has only one element and the map is not a bijection.\n",
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