Classifying space

Remarks

Any two classifying spaces of $G$ are homotopy equivalent, so $BG$ is well defined up to homotopy. For a CW complex $X$, isomorphism classes of principal $G$-bundles over $X$ correspond to homotopy classes of maps $X \to BG$. A principal $G$-bundle over a CW complex with contractible total space is universal, so its base is a $BG$.

Example: the circle group $S^1$ acts freely on the unit sphere $S^\infty$ of $\C^\infty$, with quotient $\C P^\infty$. Since $S^\infty$ is contractible, $\C P^\infty$ is a classifying space of $S^1$.

Non-example: $\C P^1$ is not a classifying space of $S^1$. The 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 $\pi_3(\C P^\infty) = 0$.

For a discrete group $G$, a classifying space is the same as a classifying space of the group, an Eilenberg–MacLane space $K(G, 1)$. Other examples: $BO(n)$ and $BU(n)$ are infinite Grassmannians, and $B\Z = S^1$.