TITLE: Topology on $p$-adic period rings in an article by Fontaine, part II
QUESTION [6 upvotes]: This is a follow-up to this question. See that question for background and relevant notation.    
In the answer to that question, it is claimed, if I understand the answer correctly, that a basis of neighborhoods of $W(R) \otimes_{W(k)} K = W(R)[1/p]$ is given by sets of the forms $p^{-n}W(\mathfrak{a})+p^NW(R)$ for $\mathfrak{a}$ a non-zero ideal of $R.$  On the other hand, if I consider the following notes, pg. 65, Exercise 4.5, it is claimed that a basis of neighborhoods of $W(R)[1/p]$ is given by sets of the form $$U_{N,\mathfrak{a}} = \bigcup_{n > -N}  p^{-n}W(\mathfrak{a}^{p^n})+p^NW(R), N \geq 0,$$ where $\mathfrak{a}$ is a non-zero ideal.  
Is it true that the two topologies thus defined are the same? If so, why is this the case? It is easy to see that each $U_{N,A}$ contains a basis element of the form $p^{-n}W(A)+p^NW(R),$ but the other inclusion is not clear to me (is it even true?).

REPLY [6 votes]: Let me just make a small comment that while the question may seem subtle from the perspective of topological rings, from the perspective of condensed rings everything really is "obvious".  The reduction $\mathcal{O}_{\mathbb{C}_p}/p$ is discrete, so there's no question there.  Then you just place that discrete ring in the condensed world, and do all the constructions in there.  So form the inverse limit over Frobenius $\mathcal{O}_{\mathbb{C}_p}^\flat$, pass to Witt vectors, then invert $p$. This gives a condensed ring: there's nothing to check.  On $T$-valued points for a profinite set $T$, it's the analogous period ring construction for the perfectoid algebra $Cont(T;\mathbb{C}_p)$ replacing $\mathbb{C}_p$.  From this perspective, the condensed or topological structure on period rings is just a shadow of their structure as a pro-etale sheaf on perfectoid spaces, as exploited in Scholze's old (?!) article on $p$-adic Hodge theory of rigid analytic spaces.
Now, there still remains the question of comparing this condensed structure to the topological structure in Fontaine's definition.  Recall that there is a fully faithful functor, $X \mapsto \underline{X}$ with $\underline{X}(T)= Cont(T,X)$, from compactly generated weak Hausdorff topological spaces to condensed sets; this passes to rings as well.  The Witt vectors of $\mathcal{O}_{\mathbb{C}_p}^\flat$ is a countable inverse limit of discrete rings, so it is a sequential Hausdorff space and hence compactly generated.  Now it suffices to note that multiplication by $p$ map is a closed inclusion (the quotient is $\mathcal{O}_{\mathbb{C}_p}^\flat$ as the latter is perfect), and that the fully faithful functor $X \mapsto \underline{X}$ above sends sequential unions along closed inclusions to sequential unions, by the standard fact in point-set topology that if a compact Hausdorff space is a sequential union of closed subpsaces, then it is equal to one of those closed subspaces.
However, probably in general the topological structure is not so important once you have the condensed structure.  So one should probably just be happy with the first paragraph of this answer.