0

If we say that a metric space $(\mathscr{X}_1, \rho_1)$ is a subset of another metric space $(\mathscr{X}_2, \rho_2)$ if $\exists \tilde{\mathscr{X}_2} \subset \mathscr{X}_2$ s.t. $(\mathscr{X}_1, \rho_1)$ and $(\tilde{\mathscr{X}_2}, \rho_2)$ are isometrically isomorphic (we also say $(\mathscr{X}_1, \rho_1)$ is embedded in $(\mathscr{X}_2, \rho_2)$) and denote simply $(\mathscr{X}_1, \rho_1) \subset (\mathscr{X}_2, \rho_2)$. I wonder if $(\mathscr{X}_1, \rho_1) \subset (\mathscr{X}_2, \rho_2)$ and $(\mathscr{X}_2, \rho_2) \subset (\mathscr{X}_1, \rho_1)$, then whether $(\mathscr{X}_1, \rho_1)$ and $(\mathscr{X}_2, \rho_2)$ are isometrically isomorphic or not? I suppose that's true but I couldn't give a rigorous proof.

If the answer is no, then is there any condition on which the proposition is true?

xdyy
  • 141
  • This is answered here: https://mathoverflow.net/questions/73309/isometries-between-metric-spaces In brief, it's false for arbitrary metric spaces but true for compact metric spaces. – Daniel Hast Mar 05 '22 at 13:58
  • @DanielHast Thanks so much. I'll go check it. – xdyy Mar 05 '22 at 13:59

0 Answers0