My advisor suggested me to study complete local rings and said I should insist on things like the $\mathfrak{m}$-adic topology on a complete local ring $(R,\mathfrak{m})$. Unfortunately, in the books he recommended to me I couldn't find any such statement so I tried to prove it myself.
The only idea I had (based by intuition) was to show some $\mathfrak{m}^j$ was compact. By a direct proof showed that my intuition may be bad, as I cannot always choose a finite subcover of $\{x+\mathfrak{m}^{j+1}\}_{x\in\mathfrak{m}^j}$. So I asked my advisor at least for some hint, but he didn't respond to my message.
Subsequently, I found a proof for complete DVRs, but they required that the residue field $R/\mathfrak{m}$ to be finite.
So my question is: is it always true that a complete local ring is locally compact? If so, how can it be proven?
Not at all, locally compact would mean some $m^l$ would be compact, equivalently $m^l/ m^s$ should be finite for all $s \ge l$. Take any field $k$ and $R = k[[t]]$ the ring of power series in $t$ with coefficients in $k$. The problem is that $\dim_k (m^a/m^b) = b-a$. But $k$ might be infinite...