Classifying space for SO(n)

In mathematics, the classifying space ${\displaystyle \operatorname {BSO} (n)}$ for the special orthogonal group ${\displaystyle \operatorname {SO} (n)}$ is the base space of the universal ${\displaystyle \operatorname {SO} (n)}$ principal bundle ${\displaystyle \operatorname {ESO} (n)\rightarrow \operatorname {BSO} (n)}$. This means that ${\displaystyle \operatorname {SO} (n)}$ principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into ${\displaystyle \operatorname {BSO} (n)}$. The isomorphism is given by pullback.