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Let $G$ be a topological group and $X$ be a topological space with $G$-action. Suppose there are two topologies $A$ and $B$ over $X$, with topology $A$ finer than $B$.

Question: Does $G$-action over $X$ being continuous (i.e. $\varphi: G\times X \to X$ being continuous) w.r.t topology $A$ imply it is also continuous w.r.t topology $B$?

For example is it true that: if $G$ acts continuous over $X$ for the discrete topology, then it is continuous for any other topology?

Remark.

1.For a given continuous map $f:X\to Y$, I know that it is still continuous when we replace topology of $X$ with a finer one and the topology of $Y$ with a coarser one. I didn't see why my question can be more subtle than this general case, but I have impression that this is actually true (but I can't come up with an easy argument, nor can I show this is wrong by some easy examples).

2.I know that when $X$ is endowed with the discrete topology, then action $G$ is continuous if the stabilizer $Stab_{G}(x)$ of any point $x\in X$ is open in $G$. I didn't see how can this help to show that $\varphi^{-1}(U)$ is open in $G \times X$ for any open $U$ (in another topology of $X$). I have $\varphi^{-1}(U)\supset \bigcup\limits_{x\in U}x\times Stab_{G}(x)$, but this is not enough to conclude.

Thanks for the nice counter-example given below by @Aryaman Maithani, the question hence has negative answer in general. I changed a little the question, cf. Continuity of Galois action under changing the topology of G-set to a coarser one.

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    $\begingroup$ Welcome to Mathematics SE. Take a tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. $\endgroup$
    – Shaun
    Jul 15, 2021 at 20:45
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    $\begingroup$ Thanks @Shaun ! I explained a little more. $\endgroup$
    – Basic
    Jul 15, 2021 at 20:52
  • $\begingroup$ You're welcome, @Basic. Your title could use some work, too. I suppose you're in a better position to summarise the question than I am. $\endgroup$
    – Shaun
    Jul 15, 2021 at 21:38
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    $\begingroup$ I modified the title to be more precise, thanks again for the suggestion! @Shaun $\endgroup$
    – Basic
    Jul 15, 2021 at 21:50

1 Answer 1

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Consider $G = \{-1, 1\}$ (group under multiplication) with discrete topology and $X = \Bbb R$ and the action given as $(g, x) \mapsto gx$.

If $X$ is the given the discrete topology, then the action is continuous. (The product $G \times X$ is also discrete.)

If $X$ is given the lower limit topology, then the action is not continuous. (The preimage of $[0, 1)$, for example, is not open.)

If $X$ is given Euclidean topology, then again it is continuous. (Check preimage of $(a, b)$.)

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    $\begingroup$ Thanks for the nice counter-example! I see the question doesn't make too much sense allowing $G$ to take arbitrary topology. I now edit the question, fixing the topology of $G$ and only change the topology of $X$. $\endgroup$
    – Basic
    Jul 16, 2021 at 17:41
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    $\begingroup$ @Basic: I would suggest you ask that question separately. Add a link to this post showing how the original statement doesn't hold in full generality. One benefit is that more people will see the question that you actually want to ask. Second is that it's considered better to restrict to only one question in a post. $\endgroup$
    – Aryaman Maithani
    Jul 16, 2021 at 17:53
  • $\begingroup$ @Basic: One more thing to notice is that my $G$ is $\operatorname{Gal}(\bar{K}/K)$ for $K = \Bbb R$. Of course, $\Bbb R$ is not a number field and so it does not actually solve your problem but an observation nonetheless. $\endgroup$
    – Aryaman Maithani
    Jul 18, 2021 at 9:38
  • $\begingroup$ Thanks @Aryaman Maithani for the remarks. I will modify the questions. $\endgroup$
    – Basic
    Jul 18, 2021 at 16:55

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