For questions on vector bundles, a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$.

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$ (for example $X$ could be a topological space, a manifold, or an algebraic variety): to every point $x$ of the space $X$ we associate (or "attach") a vector space $V(x)$ in such a way that these vector spaces fit together to form another space of the same kind as $X$ (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over $X$. Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles.