Classification of real line bundles (${\mathrm{Vect}}_{\mathbb{R}}^1(X)\cong H^1(X;{\mathbb{F}}_2)$)

Real line bundles over a paracompact space are classified up to isomorphism by their first Stiefel–Whitney class in $H^1(X;{\mathbb{F}}_2)$. Tensor product corresponds to addition of classes.