Bundle maps, sections, subbundles, and isomorphisms

For vector bundles $p:E\to X$ and $p^{\mathrm{\prime }}:E^{\mathrm{\prime }}\to Y$, a vector-bundle map over $f:X\to Y$ is a continuous map $T:E\to E^{\mathrm{\prime }}$ satisfying $p^{\mathrm{\prime }}T=fp$ whose restriction $E_x\to E_{f(x)}^{\mathrm{\prime }}$ is linear for every $x$. A bundle isomorphism is an invertible bundle map over the identity of the base.

A section is a continuous $s:X\to E$ with $ps={\mathrm{id}}_X$. It is nowhere zero if $s(x)\mathrm{\ne }{0}_x$ for all $x$.

A subset $E^{\mathrm{\prime }}\subseteq E$ is a rank-$r$ vector subbundle when each $E_x^{\mathrm{\prime }}$ is an $r$-dimensional linear subspace and every point has a bundle chart carrying $E^{\mathrm{\prime }}{\mathrm{\mid }}_U$ to $U\times ({\mathbb{F}}^r\times \{0\})\subseteq U\times {\mathbb{F}}^n$. Thus constant fiber dimension alone does not replace local triviality.

A sequence $0⟶E^{\mathrm{\prime }}\stackrel{i}{\to }E\stackrel{q}{\to }{E}^{\mathrm{\prime }\mathrm{\prime }}⟶0$ is short exact over $X$ when its maps lie over ${\mathrm{id}}_X$, the sequence on each fiber is exact, and the kernel and image have the stated subbundle structures. These conventions refine Real and complex topological vector bundles.