I apologize in advance that this question is likely ridiculously easy.
If $A$ is a topological ring and $A_0, A_1$ are open subrings, is the subring $A_0A_1$ open?
If this is not true, then the precise statement I'd like to understand is given in corollary 1.3 i) of
The subring $A_0A_1$ is defined as the set of sums of products of elements in $A_0$ and elements in $A_1$ (see the sentence right before the Definition on page 456). In particular, for any $x\in A_0A_1$ and any $a\in A_0$, $x+a=x+a\cdot 1$ is also in $A_0A_1$. Since $A_0$ contains a neighborhood of $0$, this means $A_0A_1$ is open.