Classifying space for finite-dimensional torus

Question

Note that $K({\bf Z},1)=S^1$ but $BS^1 = {\bf CP}^\infty$.

For finite groups $H$, $G$, $$K(G\times H,1) = K(G,1)\times K(H,1)$$

Does it works for classifying spaces of continuous groups ?

As far as I know, $$BT^d\neq \prod_{i=1}^d {\bf CP}^\infty$$

What is $BT^d$ ?

And how can we prove $b_{2i} (BT^d, {\bf Q})=d^i$ ?

Thank you in advance.

Answer 1

$EG\times EH$ is a contractible space with free $G\times H$-action. The quotient by this action is $BG\times BH$. So $B(G\times H)\cong BG\times BH$ (in particular, $B(S^1)^d\cong\mathbb(CP^\infty)^d$).