Stable Grassmannian

Definition 9.2.20. We define the stable Grassmannian \[ BU := \colim \left (\Gr _0(\C ^\infty ) \xrightarrow {V \mapsto V \oplus \C } \Gr _1(\C ^\infty ) \xrightarrow {V \mapsto V \oplus \C } \Gr _2(\C ^\infty ) \to \cdots \right ), \] where the transition maps are induced by the standard isometric identification \(\C ^\infty \oplus \C \cong \C ^\infty \). The Schubert cell structures make these transition maps inclusions of subcomplexes, so their colimit \(BU\) is again a cell complex; see Hatcher (2003), p. 34. In particular, the transition maps are closed embeddings, and we equip \(BU\) with the colimit topology.