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Assume that $(X,d)$ is a separable complete metric space a denote by $G$ the group of all isometries of $(X,d)$. It is well-known that $G$ equipped with the pointwise convergence topology is a Polish group. Moreover, this group naturally acts on $X$ by $g \cdot x = g(x)$ and this action is continuous.

Now, given a point $x \in X$, I would like to know what can be said about the orbit (with respect to the action described above) of the point $x$ from the viewpoint of descriptive set theory. In other words, I would like to know the descriptive quality of the set $\{ g(x)\, ; \ g \in G \}$. Clearly, this set is analytic, but I would actually like it to be closed :-)

I will appreciate any help.

Jenda358
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  • +1 I like this question. – GEdgar Dec 16 '20 at 15:33
  • Let $G\curvearrowright X$ be a Borel action of the Polish group $G$ on a Polish $X$. Then point stabilizers are closed and orbits are analytic, but they don't have to be closed or particularly nice. There are results along those lines, for example $Gx$ is $G_\delta$ iff $Gx$ is nonmeager in the induced topology, iff $gG_x\to Gx$ is an homeomorphism of $G/G_x$ onto $Gx$ – Alessandro Codenotti Dec 16 '20 at 18:00
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    People are usually interested in studying the equivalence relation $E^X_G\subseteq X^2$ with $xE^X_Gy$ iff they are in the same orbit. The relation $E^X_G$ is always analytic but in general not even Borel. Entire books have been written on similar topics (the area is called invariant descriptive set theory), see for example works of Kechris or the book on orbit equivalence by Hjorth – Alessandro Codenotti Dec 16 '20 at 18:02
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    (Of course the situation might be nicer when $G=\mathrm{Iso}(X)$, but I doubt it even though I don't know exactly how bad the situation is, which is why I'm writing comments instead!) – Alessandro Codenotti Dec 16 '20 at 18:06

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