In 'Theory of p-adic Galois representations' (Fontaine-Ouyang), given a topological commutative ring B equipped with a continuous compatible action of a topological group G the authors define: A $\textbf{B-representation}\; X$ of $G$ to be a $B$-module of finite type equipped with a continuous and semi-linear action of $G$.
My question is when we are talking about the continuous action of $G$ on $X$ are we giving $X$ the quotient topology it inherits from the surjection $B^d\rightarrow X$ for some $d$, where $B^d$ is given the product topology. If so, then is it obvious that the topology does not depend on the choice of the surjection?
Similarly when we tensor $B$ representations, I don't see a natural way of equipping it with a topology, in case $B$ is a field, then do we equip it with the topology of $B^n$ for example?
I don't know whether I am missing something obvious or if this is a stupid question.