Classifying space for O(n)

In topology, a classifying space for a topological group provides a way to classify principal bundles associated with that group. For the orthogonal group \( O(n) \), the classifying space is denoted \( BO(n) \). ### Understanding \( BO(n) \): 1. **Definition**: The classifying space \( BO(n) \) is defined as the space of all oriented real n-dimensional vector bundles.