Is the coherent topology well behaved with respect to products? Let $$A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots$$ be a nested sequence of topological spaces such that each inclusion is continuous and each $A_i$ is closed in $A_{i+1}$. Now declare $X\subseteq \mathcal{A} := \bigcup_i A_i$ to be open if and only if $X \cap A_i$ is open for every $i$.  This is called the coherent or "final" topology associated to the filtration of $\mathcal{A}$. 
My Question: If $\mathcal{B}$ is a topological space, is the product topology on $\mathcal{A} \times \mathcal{B}$ the same as the coherent topology with respect to the filtration $$A_1 \times \mathcal{B} \subseteq A_2 \times \mathcal{B} \subseteq A_3 \times \mathcal{B} \subseteq \cdots?$$
If not, are there conditions on $\mathcal{B}$ for which the two coincide?
Motivation:
Each Stiefel Variety $V(k,n)$ is a principal $O(k)$ bundle over the Grassmanian $G(k,n)$. The Stiefel Varieties are nested, as above and we want their union, $V(k, \infty)$ (with the coherent topology) to be a principal $O(k)$ bundle over $G(k, \infty)$.  In particular we must have the continuity of the group action $O(k) \times V(k, \infty) \rightarrow V(k, \infty)$.  This action restricts to a continuous action on each $V(k, n)$ and so continuity will follow if the topology of $O(k) \times V(k, \infty)$ is coherent with the subspaces $O(k) \times V(k, n)$.
 A: One way to define $\mathcal A$ it is to notice that $\mathcal A = \varinjlim A_i$, the direct limit of the system $(A_i)$ in the category of topological spaces. It is characterized by the fact that
$$\hom(\varinjlim A_i, C) = \varprojlim \hom(A_i, C)$$
for any topological space $C$.
Now, we would like this direct limit to commute with a product. One situation in which this happens is if the product admits a right adjoint. This is true, for instance, in the category of compactly generated spaces, where we have
$$\hom(A\times B, C) \simeq \hom(A, C^B)$$
where $C^B = \hom(B, C)$ with the compact-open topology. In this case, we have the sequence of canonical bijections
$$\begin{eqnarray}\hom((\varinjlim A_i) \times B, C) &=& \hom(\varinjlim A_i , C^B)\\&=&\varprojlim \hom(A_i, C^B) \\&=& \varprojlim \hom(A_i \times B, C) \\&=& \hom(\varinjlim (A_i \times B), C).\end{eqnarray}$$
Since this is true for any $C$, the Yoneda lemma implies that $(\varinjlim A_i) \times B \simeq \varinjlim (A_i \times B)$.
Of course, this is just the proof that a left adjoint functor commutes with direct limits.
