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  "body": "\n## Summary\n\nThe classifying space $BG$ of a topological group $G$ is the base of a [[universal bundle|1HDH716Z24]] $EG \\to BG$.\nPrincipal $G$-bundles over a CW complex $X$ correspond to homotopy classes of maps $X \\to BG$.\nCohomology classes of $BG$ give the [[characteristic classes|7HACJC0JE2]] of such bundles.\nFor a discrete group $G$, the space $BG$ is the [[classifying space of the group|WA796ZCNRX]] in the sense of an Eilenberg–MacLane space $K(G, 1)$.\n\n## Explanations\n\n### L1 human\n\nA classifying space is a catalogue of all bundles with a given structure group.\nEach map from $X$ into the catalogue picks out a bundle over $X$, and deforming the map does not change the bundle.\nFor real line bundles the catalogue is the infinite projective space $\\R P^\\infty$, the space of lines through the origin in $\\R^\\infty$.\nMapping a point of $X$ to a line tells which line sits over that point.\n",
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