Vector bundle class functor is homotopy-invariant for paracompact

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 $A$ and $B$ are paracompact Hausdorff spaces and $f_{0}, f_{1} : A \to B$ are continuous maps. Then, the functorially induced maps by the vector bundle class functor, namely:

$V e c t^{n} (f_{0}) : V e c t^{n} (B) \to V e c t^{n} (A)$

and:

$V e c t^{n} (f_{1}) : V e c t^{n} (B) \to V e c t^{n} (A)$

are equal, i.e. $V e c t^{n} (f_{0}) = V e c t^{n} (f_{1})$ .

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.