Vtyjkyuk

So this is a sumamry of the construction given in pg14 of Hatchers of the tensor of two vector bundles:

Let $p_1:E _1 rightarrow B$, and $p_2 : E_2 rightarrow B$. We define $E_1 otimes E_2$ to be the disjoint union of vector spaces $p^-1(chi) otimes p_2^-1(chi)$ for $chi in B$ with the following topology:

We take an open cover of $B$ over which both initial bundles are trivial. Choose hoemomoprhisms,
$$h_i:p_i^-1(U) rightarrow U times mathbbR^n_i$$
for each open set $U$ in such a cover. The topology on $tau_U$ on the set $p_1^-1(U) otimes p_2^-1(U)$ is defined by letting the tensor product map
$$h_1 otimes h_2: p^-1_1(U) otimes p_2^-1(U) rightarrow U times (mathbbR^n_1 otimes mathbbR^n_2)$$

I am just confused with what exactly is meant here. So then what is the topology on $E_1 otimes E_2$? I suppose it is the colimit topology from the open cover.

You are right. That is the usual procedure to topologize a "bundle" over a base space $B$ which is given by associating to each $b in B$ a vector space $E_b$. Define $E= bigcup_b in B b times E_b$ = disjoint union of the fibers $E_b$ and $p : E to B, p(b,e_b) = b$. Assume you have an open cover $mathcalU$ of $B$ and for each $U in mathcalU$ an "algebraic bundle isomorphism" $h_U : p^-1(U) to U times mathbbR^n$ (that is, $h_U(b,e_b) = (b, phi_U^b(e_b))$ with linear isomorphism $phi_U^b : E_b to mathbbR^n$, $b in U$). If the obvious compatibility conditions are satisfied (the transition functions between the "bundle charts" $h_U$ have to be homeomorphisms), then you can topologize $E$ by

(1) giving each $p^-1(U)$ the unique topology making $h_U$ a homeomorphism

(2) giving $E$ the colimit topology from the cover $p^-1(U)$, $U in mathcalU$.

In your case the situation is that any two (topological) bundle charts $h_i : p_i^-1(U) to U times mathbbR^n_i$ provide an algebraic bundle isomorphism $h_1 otimes h_2: p^-1_1(U) otimes p_2^-1(U) rightarrow U times (mathbbR^n_1 otimes mathbbR^n_2)$. You have to check compatibility and you are done.