I want to know if this notion of completed tensor product is the one that yields $$ k[\![ x ]\!] \hat{\otimes} k[\![ y ]\!] \cong k[\![ x,y ]\!]. $$ Here I should be considering the inverse limit topology in the power series rings, and $R=k$ a field (with the trivial topology?). If it is not, then what is the right notion of $ \hat{\otimes}$ ?
I believe this is broadly used when talking of formal groups laws (eg defining its contravariant bialgebra).