Characteristic class

Definition

A characteristic class is a function that associates, to every principal $G$-bundle $P\to X$ an element $c(P)$ in the cohomology algebra $H^{*}(X)$, such that if $f:Y\to X$ is a continuous map, then $c(f^{*}P)=f^{*}(cP)$ where the $f^{*}$ on the left is the usual pullback of the bundle, and the $f^{*}$ on the right is the induced map on the cohomology.