Real Grassmannian analogues

Remark 9.2.22. The Grassmannian and Stiefel-manifold results above have real analogues with \(\C \) replaced by \(\R \) and \(U(k)\) by \(O(k)\), with the same proofs. In particular, \(\Gr _k(\R ^\infty )\) classifies rank \(k\) real vector bundles over paracompact Hausdorff spaces, \(V_k(\R ^\infty )\) is contractible, and \(\Pi _\infty \Gr _k(\R ^\infty )\simeq \bB O(k)\). We will use these facts in Chapter 10.