The functor $k_{U(d)}$ of the set of isomorphism classes of numerable principal $U(d)$-bundles is represented by $BU(d)$ the Milnor construction of the classifying space. The restriction of $k_{U(d)}$ to the full subcategory of paracompact Hausdorff spaces is represented by the Grassmannian $Gr_d$. However, it seems the Milnor construction $BU(d)$ is also paracompact Hausdorff (see this MO question), so it should be homotopy equivalent to $Gr_d$.
Is there any wrong in the above argument? If it is true, it holds $k_{U(d)}\cong[-,Gr_d]$, then how to give a classifying map to $Gr_d$ from a numerable $U(d)$-bundle over a general space, WITHOUT a COUNTABLE numerable local trivialization?
