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A continuously varying family of one-dimensional vector spaces over a topological space. A related tag is the vector-bundles tag.

A real line bundle on a topological space $$X$$ is given by a topological space $$E$$ and a continuous surjective map $$\pi : E \to X$$ such that

• $$\pi^{-1}(x)$$ has the structure of a one-dimensional vector space over $$\mathbb{R}$$,
• each point of $$X$$ has an open neighbourhood $$U$$ and a homeomorphism $$\varphi : U\times\mathbb{R} \to \pi^{-1}(U)$$ satisfying

• $$\varphi((x, v)) \in \pi^{-1}(x)$$, and
• $$v \mapsto \varphi((x, v))$$ is an isomorphism from $$\mathbb{R}$$ to $$\pi^{-1}(x)$$.

A complex line bundle is defined similarly by replacing $$\mathbb{R}$$ by $$\mathbb{C}$$ and requiring the isomorphism $$v \mapsto \varphi((x, v))$$ to be an isomorphism of complex vector spaces.

In the case that $$X$$ is a smooth manifold, one can also define the notion of a smooth line bundle, and if $$X$$ is complex, there is a notion of a holomorphic line bundle.

Line bundles are rank one vector bundles.