# Vector bundle class functor is homotopy-invariant for paracompact

From Topospaces

*This article describes a result applicable for real vector bundles over a paracompact Hausdorff space. In particular, the result is applicable for real vector bundles over a manifold, CW-space, or metrizable space*

## Statement

Suppose and are paracompact Hausdorff spaces and are continuous maps. Then, the functorially induced maps by the vector bundle class functor, namely:

and:

are equal, i.e. .

## Corollaries

- Any homotopy equivalence of paracompact Hausdorff spaces induces an isomorphism on the vector bundle classes over them.
- In particular, any vector bundle over a contractible paracompact Hausdorff space is trivial.