Vector bundle class functor

Definition

The vector bundle class functor of dimension $n$, denoted $Vec{t}^n$, 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

Facts

For paracompact Hausdorff spaces

Further information: Vector bundle class functor is homotopy-invariant for paracompact

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 $Vec{t}^n$ are equal.