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I am currently reading Katok's Fuchsian Groups and I am trying to understand the proof of the following theorem:

Theorem 2.2.1. $G$ acts properly and discontinuously on $X$ if and only if each point $x \in X$ has a neighborhood $V$ such that $$T(V) \cap V \neq \emptyset$$ for only finitely many $T \in G.$

Notes:

  • $X$ is a metric space.

  • $G$ is a group of isometries.

  • There's a similar question here , but the following question is not the same as in the post mentioned.

My doubt is the following: What does Katok mean when she say $$T(V) \cap V \neq \emptyset ?$$ For example, if $T(V)$ and $V$ were disjoint, it means that:

  1. Actually $T(V)$ and $V$ are disjoint?

  2. $T(V)$ and $V$ has not points of $Gx$ in common?

Assuming option 1 I couldn't follow the proof, specially in the assertion "$T(V) \cap V \neq \emptyset $ implies that $T \in G_x$ (the stabilizer of $x$ )" (you can see the beginning of the proof here ).

So I should assume the option 2?

If not, please could you explain me why this assertion is true? (preferible without mention topological facts, I have no taken Topology yet).

Thanks in advance. Any misunderstanding or definition you could require let me know.

rowcol
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    Basic topology is a large prerequisite for studying Fuchsian groups, I strongly suggest you learn some first. Your questions themselves are common sense issues in set theory, notation, and basic English. The mathematical sentence "$T(V) \cap V = \emptyset$" is equivalent to the sentence "$T(V)$ and $V$ are disjoint", which means that there does not exist anything which is an element of both of the sets $T(V)$ and $V$. Adding the word "actually" to the sentence "$T(V)$ and $V$ are disjoint" does not alter its meaning. – Lee Mosher Dec 16 '19 at 18:39
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    The metric space $X$ and the group $G$ are disjoint from each other, and $T(V)$ and $V$ are both subsets of $X$, so no element of $T(V)$ nor of $V$ can be an element of $G$. – Lee Mosher Dec 16 '19 at 18:39
  • @LeeMosher Sorry, it was a typo. I should write $Gx$ instead only $G$. I know what disjoint means in terms of empty set. The problem is that if you picture $V$ and $T(V)$ as open balls (assume $x$ is the only element of $Gx$ that is in $V$ and $T(x)$ is the only element of $Gx$ that is in $T(V)$) these balls could intercept in such way that $x$ and $T(x)$ were outside that intersection. Then the disjointness still being true. But in that case, $T(x)≠x$, so $T \notin G_x$. (Katok asserts that $T \in G_x$) – rowcol Dec 16 '19 at 19:00
  • If $V$ and $T(V)$ are open balls that intersect then they are not disjoint, regardless of whether $x$ is or is not in that intersection, or whether $T(x)$ is or is not in that intersection. What it means for $V$ and $T(V)$ to not be disjoint that there exists some point $y \in X$ such that $y \in V$ and $y \in T(V)$. – Lee Mosher Dec 16 '19 at 19:06
  • By the way, let me recommend Munkres book "Topology" for two purposes. One, of course, is for the topology prerequisites that you need to understand Fuchsian groups. But another reason is that before he starts writing about topology, he has an excellent preface all about concepts of set theory, which you might benefit from. – Lee Mosher Dec 16 '19 at 19:08
  • Sorry, disjointness still NOT being true is that I should write. – rowcol Dec 16 '19 at 19:12
  • @LeeMosher In resume: Two balls containing respectively only one element of $Gx$ could intersect in such way that these elements were outside the intersection. Then te condition $T(x)=x$ is false. And the author asserts that $T(x)=x$ – rowcol Dec 16 '19 at 19:16
  • @LeeMosher But if we write for example $T(V) \cap V = \emptyset$ to refer that $V$ and $T(V)$ has not elements of $Gx$ in common, the assertion $T(x)=x$ is true. Hence, $T(V) \cap V \neq \emptyset $ sounds to me that $T(V)$ and $V$ intersects at least in one element of $Gx$ – rowcol Dec 16 '19 at 19:21
  • It sounds like you want to redefine the basic concepts of set theory, including the emptyset $\emptyset$ and the concept of intersection $\cap$, in which case I don't think even Munkres book will be of much help to you. – Lee Mosher Dec 16 '19 at 19:22
  • @LeeMosher No, I don't want to redefine the concept. I am only confussed about what the author refers, because if I take literally the definition of empty intersection, the result she said could be not true. – rowcol Dec 16 '19 at 19:25

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