Understanding proof about properly and discontinuous action in a group

Question

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.