= Real line bundle
{title2=$L\to X$}
A <real line bundle> is a rank-one real <vector bundle>. Over a <paracompact> <Hausdorff space>, or over a <smooth manifold> in the smooth category, it is orientable exactly when it is a <trivial vector bundle>.
To prove the nontrivial implication, choose a <fiber metric>. An <orientation of a vector bundle> selects one of the two unit <vectors> in each fiber. These choices agree continuously in oriented <local frames>, giving a global <nowhere-zero section>. The <map> $(x,t)\mapsto t\,s(x)$ is then a fiber-linear product <isomorphism>. Conversely, a <trivial vector bundle> has the positive standard <basis> as a continuous <orientation of a vector bundle>.
The <Möbius line bundle> over a <circle> is the basic counterexample to triviality: one circuit reverses the fiber orientation, so no global positive unit <section of a vector bundle> can be chosen.
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