the infinite Grassmannian
/ GRASS-muh-nee-un /
A finite Grassmannian Gr(k, n) is the space of all k-dimensional subspaces (planes through the origin) sitting inside R^n: a point of it is literally a flat k-plane. The infinite Grassmannian is what you get by letting the ambient dimension run to infinity, Gr(k, infinity) = colim_n Gr(k, n), the space of all k-planes in the infinite-dimensional R-infinity (or C-infinity in the complex case). It is roomy enough to hold a copy of every k-plane you could ever need.
Over the Grassmannian there sits a tautological bundle, the most honest bundle imaginable: over the point representing a k-plane P, the fiber is P itself. This tautological bundle is the universal bundle, and the infinite Grassmannian is therefore a concrete model of the classifying space — Gr(n, C-infinity) realizes BU(n), and the real version realizes BO(n). To classify rank-n bundles you map X into Gr(n, infinity) by, at each point x, naming the n-plane that is the fiber of your bundle there (after embedding the fibers into a common big space using a partition of unity), and pulling back the tautological bundle recovers yours.
This makes the classification theorem geometric and almost tautological in spirit: a bundle is a way of choosing a plane over each point, and the Grassmannian is the universal space of planes. Its cohomology is computed by Schubert cells, giving H^*(Gr(n,C-inf)) = Z[c_1,...,c_n] with the Chern classes as generators — so the Grassmannian is simultaneously where bundles are classified and where characteristic classes are born. The caveat is dimension bookkeeping: you must let the ambient space be infinite-dimensional, because a fixed finite R^N can only embed bundles up to a rank/dimension bound, and the universal property needs all of them.
The space of 1-dimensional subspaces (lines) of C-infinity is Gr(1, C-infinity) = CP-infinity, the infinite complex projective space. Its tautological line bundle is the universal complex line bundle, and pulling it back along a map X -> CP-infinity gives every complex line bundle on X. This is the same statement as BU(1) = CP-infinity from the classifying-space picture, now seen as a Grassmannian.
CP-infinity is both the infinite Grassmannian of lines and the classifying space BU(1).
The ambient dimension must be infinite: over a finite Gr(n, N) every bundle of low enough rank is captured, but the strict universal property — every bundle, no dimension cap — needs the colimit. Beginners sometimes try a fixed finite N and lose bundles of high cell-dimension.