TITLE: The Stiefel–Whitney classes of a Cartesian product
QUESTION [9 upvotes]: Exercise 4-A of Milnor and Stasheff's book Characteristic Classes reads:

Show that the Stiefel–Whitney classes of a Cartesian product are given by $$w_k(\xi\times\eta) = \sum^k_{i=0} w_i(\xi)\times w_{k-i}(\eta)$$

This does not make sense to me because if $\xi\rightarrow B_1$ and $\eta\rightarrow B_2$ then $\xi\times\eta\rightarrow B_1\times B_2$, so $w_k(\xi\times\eta)\in H^k(B_1\times B_2;\mathbb{Z}_2)$. While $w_i(\xi)\times w_{k-i}(\eta)=(w_i(\xi),w_{k-i}(\eta))\in H^i(B_1;\mathbb{Z}_2)\times H^{k-i}(B_2;\mathbb{Z}_2)$.
So $w_k(\xi\times\eta)$ and $\sum^k_{i=0} w_i(\xi)\times w_{k-i}(\eta)$ lie in different sets, so how can I be asked to prove that they are equal?

REPLY [6 votes]: I think you did not understand the $\times$ symbol at here. As others pointed out it is the cross product. So you have
$$
w_{k}(\epsilon\times \delta)=w_{k}(\pi_{1}^{*}(\epsilon)\oplus\pi_{2}^{*}(\delta))
$$
By axiom 4 on page 38 we have
$$
w_{k}(\pi_{1}^{*}(\epsilon)\oplus\pi_{2}^{*}(\delta))=\sum^{k}_{i=1}w_{i}(\pi_{1}^{*}(\epsilon))\cup w_{k-i}(\pi_{2}^{*}(\delta))
$$
While by the definition of cross product we have
$$a\times b=\pi_{1}^{*}(a)\cup \pi_{2}^{*}(b)$$
This proved the original statement. Here the definition of cross product follows from Hatcher, page 218. Note as others pointed out the original statement is a mere restatement of Kunneth theorem with base ring be a field.