Are (co)limits of topological rings in $\mathsf{Ring}$ with continuous legs always a (co)limit in $\mathsf{Top}$ for a suitable topology? This question is motivated by the characterization of the $p$-adic numbers as a limit of finite fields via truncation. 
Since my knowledge of the former is close to nonexistent, I will try to ask the question in a self-contained manner: suppose that we have a (small) diagram
$$
F : J \to \mathsf{Ring}
$$
with $Fj$ a topological ring for each $j \in J$, and that $R := \lim_J F$ has the final topology with respect to the legs
$$
\pi_j : R   \to F_j.
$$
Do we necessarily have that $R = \lim_J F$ in $\mathsf{Top}$? 
Some aditional context: I was trying to extend what we can do in $\mathsf{Set}$, namely that we can take the (co)limit there and then take final/initial topologies to obtain a (co)limit of topological spaces. Certainly we can forget structure to do this, but we have no way of ensuring that legs will be arrows in $\mathsf{Ring}$, where as in $\mathsf{Top}$ one can fix this by choosing the 'correct' topology on the (co)limit in $\mathsf{Set}$. 
So, one approach would be to take the limit in $\mathsf{Ring}$ and then take the final topology with respect to this cone. But is this a (co)limit cone of topological spaces? Or could we have found a higher/lower cone (maybe not comprised of ring morphisms).
 A: Assuming all the maps in the diagram $F$ are continuous (so that it even makes sense to talk about $\lim_J F$ in $\mathsf{Top}$), then yes, the final topology makes $R$ also the limit in $\mathsf{Top}$.  This is is just because the forgetful functor from $\mathsf{Ring}$ to $\mathsf{Set}$ preserves limits, so $R$ is also the limit in $\mathsf{Set}$, and as you mention limits in $\mathsf{Top}$ can be computed by taking the limit in $\mathsf{Set}$ and then giving it the final topology.
For colimits though this fails horribly, since the forgetful functor from $\mathsf{Ring}$ to $\mathsf{Set}$ does not preserve colimits.  For instance, if you consider the empty diagram, then its colimit in $\mathsf{Ring}$ is $\mathbb{Z}$, but its colimit in $\mathsf{Set}$ and $\mathsf{Top}$ is $\emptyset$.  If the colimit of $F$ happens to be preserved by the forgetful functor from $\mathsf{Ring}$ to $\mathsf{Set}$ (for instance, when $J$ is filtered), then it will be true that the colimit in $\mathsf{Top}$ is just $R$ with the initial topology, since colimits in $\mathsf{Top}$ can be computed by taking the colimit in $\mathsf{Set}$ and then giving it the initial topology.
