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Questions:

(1) Let $K$ be a number field and $\overline{K}$ an algebraic closure. Let $G=Gal(\overline{K}/K)$ the absolute Galois group endowed with the profinite topology, and $M$ some topological space endowed with a $G$-action. Suppose the $G$-action over $X$ is continuous w.r.t the discrete topology, does this imply it is also continuous w.r.t a coarser topology?

(2) The same question as (1), but instead we assume $K$ is a local field (a discrete valuation field with a perfect residue field).

Remarks:

(1) This question is a continuation of: Continuity of $G$-action under changing the topology of $G$-set to a coarser one.

(2) The remark given by Moishe Kohan below gives a counter-example for the question.

Basic
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    The same example of $Z/2$-action as in the answer to the other question works here too, with $K={\mathbb Q}$, since $Gal(\bar Q/Q)$ admits (continuous) epimorphisms to $Z/2$, see for instance here. – Moishe Kohan Jul 18 '21 at 17:18
  • Thanks @MoisheKohan for the remark! – Basic Jul 18 '21 at 21:17
  • @Moishe Kohan : Sorry maybe I didn't get the point. Did you mean that $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ (endowed with profinite topology) acting over $\mathbb{R}$ by factoring through $\mathbb{Z}/2$, and then acts (as in the answer of the question https://math.stackexchange.com/questions/4199556/continuity-of-g-action-under-changing-the-topology-of-g-set-to-a-coarser-one), is a counter-example? – Basic Jul 19 '21 at 12:55
  • Of course...... – Moishe Kohan Jul 19 '21 at 12:55
  • @Moishe Kohan : I see, this is a clever observation and it works, maybe also works for $K=\mathbb{Q}_p$? However, what really in my mind was to look for the situation where the question has positive answer. But by your remark, the restriction that $G$ is endowed with the profinite topology is almost useless... I don't even know how to change the question now. Anyway, thanks again for the nice remark! – Basic Jul 19 '21 at 13:16
  • $Q_p$ is not a number field and you assumed that $K$ is. I do not know of any interesting situation where your question has positive answer. – Moishe Kohan Jul 19 '21 at 13:33
  • Yes, I mean for the local field case in question (2). Thanks for the remarks. – Basic Jul 19 '21 at 14:14

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