Lets say I have an inverse limit, $\varprojlim R_{i}$ given by the an inverse system $\{(R_{i})_{i\in I},(\phi_{ij})_{i\leq j ; i,j\in I}\}$ where $I$ is our directed poset and our objects $R_i$ are rings. Then we define our inverse limit topology as the smallest topology on $\varprojlim R_{i}$ such that our natural projections $\varprojlim R_{i}\to R_i$ are continous functions. From my understanding we can construct this certain topology, by first taking the product of the $R_i$ and definining an initial topology there. We take the preimages of the open sets of each $R_i$ under the projection morphism as our subbase, ie. our subbase for our topology is $\{\pi_{i}^{-1}(U): U\subset R_i \text{ open set in } R_i \}$ where $\pi_{i}:\prod_{i\in I} R_i\to R_i $ is the natural projection. Then we just endow $\varprojlim R_{i}$ with the subspace toplogy.
My question is: if I'm reading a book or some notes and its not mentioned what topology each $R_i$ is endowed with , is it common standard to assume its the discrete topology for each $R_i$?