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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.

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