*Thank you to Edward Frenkel for kindly explaining the difference between $CP^\infty$ and $Pic(X)$ (both classifying spaces of line bundles), and to Qiaochu Yuan for explaining why on earth $CP^\infty$ is the moduli space of line bundles over a point. Any errors are mine, not theirs.*

As we saw in a Precursor to Characteristic Classes, $CP^\infty$ is the classifying space of complex line bundles over $X$.

$CP^\infty$ is, in some sense, the moduli space of line bundles over a point. There’s only one isomorphism class of line bundles over a point — but then this one line bundle has automorphism group $C^\times$ (which is homotopy equivalent to $U(1)$).

Allow me to introduce you to something that looks a LOT like $CP^\infty$. Continue reading $Pic(X)$ vs. $CP^\infty$