Let $S$ be a unit sphere in the Urysohn space $\mathbb{U}$. Is it true that any isometry $S\to S$ can be extended to an isometry $\mathbb{U}\to \mathbb{U}$?
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asked May 5, 2022 at 19:14
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This is not true, no. There is a proof in Section 4.4 of this old paper of mine ; the key fact is that if $B$ is an open unit ball in the Urysohn space $\mathbb U$, then $\mathbb U$ is isometric to $\mathbb U \setminus B$.
(the proof of the fact about extension of isometries is given for a ball in the paper, but the same argument works if one replaces "ball" by "sphere" in the proof)
