Classifying space of real line bundles

Question

I have read that $\mathbb{R}P^{\infty}$ is the classifying space of real line bundles. But I don't understand what that means. From all I know, a classifying space refers to a group. Is this meant to be the Picard Group?

Answer 1

It means that for $X$ a (reasonable) topological space, the set $[X, \mathbb{RP}^{\infty}]$ of homotopy classes of maps $X \to \mathbb{RP}^{\infty}$ can be put in natural bijection with the set of real line bundles on $X$. More generally there is a classifying space $BG$ of principal $G$-bundles for $G$ a topological group, and this one is the special case $G = GL_1(\mathbb{R})$, or equivalently (as it turns out) $G = O(1) = C_2$.