Real line bundle ($L\to X$)

A real line bundle is a rank-one real vector bundle. It is orientable exactly when it is trivial.

The Möbius line bundle over the circle iswith projection to $t$ modulo $2\pi$. It is a smooth rank-one vector bundle. A section is represented by a function $s$ satisfying $s(t+2\pi )=-s(t)$, so the intermediate value theorem forces every continuous section to vanish somewhere. Thus it has no nowhere-zero section and is not isomorphic to the trivial vector bundle of rank one.

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.

For a real line bundle, the mod-two Euler class is its top Stiefel–Whitney class, namely $w_1(L)\in H^1(X;{\mathbb{F}}_2)$. Its evaluation on a closed curve records whether parallel transport reverses the fiber orientation.