Real and complex topological vector bundles

Fix $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ and $n\ge 0$. A rank-$n$ $\mathbb{F}$-vector bundle over $X$ is a locally trivial bundle $p:E\to X$ in the sense of Locally trivial fiber bundle, together with an $\mathbb{F}$-vector-space structure on every fiber $E_x=p^{-1}(x)$, whose local charts

${\varphi }_i:p^{-1}(U_i)\stackrel{\cong }{\to }{U}_i\times {\mathbb{F}}^n$

restrict on every fiber to linear isomorphisms $E_x\cong {\mathbb{F}}^n$. Here $M_n(\mathbb{F})\cong {\mathbb{F}}^{n^2}$ has its Euclidean topology and ${\mathrm{GL}}_n(\mathbb{F})$ has the subspace topology; its underlying algebraic set is the one fixed in Invertible matrices and the general linear group ${\mathrm{GL}}_n(F)$. The transition from chart ${\varphi }_i$ to ${\varphi }_j$ is therefore

${\varphi }_j{\varphi }_i^{-1}(x,v)=(x,g_{ji}(x)v)$

for a continuous $g_{ji}:U_i\cap U_j\to {\mathrm{GL}}_n(\mathbb{F})$. Indeed, evaluating the continuous chart change at each standard basis vector gives the matrix columns continuously, and its values are invertible because the chart change is fiberwise linear. Our index convention gives

$g_{ii}=I,g_{ki}=g_{kj}{g}_{ji}\mathrm{.}$

Rank is fixed in this definition. The case $n=0$ is the bundle $X\to X$ with zero-dimensional fibers, and $X=\mathrm{\varnothing }$ is allowed.

The bundle is numerable if it has a linear trivializing cover $(U_i)$ together with a locally finite partition of unity $({\rho }_i)$ such that $\mathrm{supp}{\rho }_i\subseteq U_i$. The charts and the subordinate partition, when specified, are called a numeration.