The Whitney-sum monoid of complex vector bundles

For a compact Hausdorff space $X$, let

${\mathrm{Vect}}_{\mathbb{C}}(X)$

be the set of isomorphism classes of finite-rank complex vector bundles over $X$. Here a finite-rank bundle is allowed to have locally varying rank: it is a finite disjoint clopen decomposition $X={\coprod }_j{X}_j$ together with a fixed-rank bundle in the sense of Real and complex topological vector bundles over each $X_j$. Equivalently, its fiber-dimension function is locally constant; compactness of $X$ makes its image finite and its rank fibers clopen. Thus no single global rank is imposed, but the notion is reduced to the library's fixed-rank bundles on finitely many clopen pieces.

Define

$[E]+[F]=[E\oplus F],0=[0_X],$

where $0_X=X\times {\mathbb{C}}^0$. This is well-defined on isomorphism classes by Bundle maps, sections, subbundles, and isomorphisms and Whitney sum, tensor, dual, Hom, and exterior-power bundles, applied on the finite common refinement of the two rank decompositions. The fiberwise swap, reassociation, and zero maps are bundle isomorphisms, so Whitney sum makes ${\mathrm{Vect}}_{\mathbb{C}}(X)$ a commutative monoid.

When $X=\mathrm{\varnothing }$, every total space of a bundle over $X$ is empty. Hence there is one isomorphism class, and ${\mathrm{Vect}}_{\mathbb{C}}(\mathrm{\varnothing })$ is the one-element monoid.