# Vector bundle

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In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space ${\displaystyle X}$ (for example ${\displaystyle X}$ could be a topological space, a manifold, or an algebraic variety): to every point ${\displaystyle x}$ of the space ${\displaystyle X}$ we associate (or "attach") a vector space ${\displaystyle V(x)}$ in such a way that these vector spaces fit together to form another space of the same kind as ${\displaystyle X}$ (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over ${\displaystyle X}$.
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space ${\displaystyle V}$ such that ${\displaystyle V(x)=V}$ for all ${\displaystyle x}$ in ${\displaystyle X}$: in this case there is a copy of ${\displaystyle V}$ for each ${\displaystyle x}$ in ${\displaystyle X}$ and these copies fit together to form the vector bundle ${\displaystyle X\times V}$ over ${\displaystyle X}$. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.