Vector bundle class functor

Definition
The vector bundle class functor of dimension $$n$$, denoted $$Vect^n$$, is a contravariant functor from the category of topological spaces with continuous maps to the category of sets, such that:


 * A topological space is mapped to the set of isomorphism classes of $$n$$-dimensional real vector bundles over the topological space
 * A continuous map between topological spaces sends a vector bundle in the image space, to its pullback bundle

For paracompact Hausdorff spaces
If $$A$$ and $$B$$ are paracompact Hausdorff spaces, and $$f_0, f_1: A \to B$$ are homotopic maps from $$A$$ to $$B$$, then the functorially induced maps $$Vect^n$$ are equal.