Let $A \subset X$ and $B \subset X$ be two isometric subsets of a metric space $X$. So there is an isometry $f: A \to B$.
Say that a metric space $X$ has the *superposition property* (my terminology) if, for every pair of isometric subsets $A$, $B$, there is an isometry of $X$, $F: X \to X$, that superimposes $A$ onto $B$: $F(A) = B$, i.e. $F$ places $A$ on top of $B$.



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> Which metric spaces have this superposition property?
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Euclidean space $\mathbb{E}^d$ has this property.
But it seems the punctured plane does not: e.g. if $A$ is the point $(1,0)$ and $B$ is the point $(-2,0)$, then (I believe) there is not an isometry of the whole punctured plane that maps $A$ onto $B$.


Has this property been studied before? If so, under what name?
I am (clearly) unschooled in this area. Thanks for pointers and/or examples!



Which metric spaces have this superposition property?
If the metric space is locally compact and intrinsic, then you get only spheres, Euclidean spaces and hyperbolic spaces.
[See *Metric methods in Finsler...* by Busemann and *Sur certaines classes d'espaces...* by Tits (1955); thanks to [Linus](https://mathoverflow.net/users/37911/linus) for [the reference](https://mathoverflow.net/a/428148/)]


Without assuming local compactness, the same conclusion holds assuming local uniqueness of geodesics [See *Metric foundations of geometry. I* by Birkhoff].
Without this extra assumption you also get the so-called [universal $\mathbb{R}$-trees](https://arxiv.org/abs/math/9904133) of finite valence; they are complete, but not separable.


If the metric is not intrinsic you get discrete spaces and yet Cantor-like spaces build on them (who knows what else).


**Comments**


* In the [Urysohn universal space](https://en.wikipedia.org/wiki/Urysohn_universal_space) $\mathbb{U}$ the property holds for compact subsets; that is, any distance-preserving map $K\to\mathbb{U}$ defined on a compact subset $K\subset \mathbb{U}$ can be extended to an isometry $\mathbb{U}\leftrightarrow\mathbb{U}$.
* The real projective space is not three-point-homogeneous --- a closed geodesic contains three points on equal distance from each other, and there is an isometric three-point set that does not lie on a closed geodesic.
* See also a [related question](https://mathoverflow.net/q/429702/).


