Ultralimit - Basic Properties of Ultralimits

Basic Properties of Ultralimits

  1. If (Xn,dn) are geodesic metric spaces then is also a geodesic metric space.
  2. If (Xn,dn) are complete metric spaces then is also a complete metric space.

Actually, by construction, the limit space is always complete, even when (Xn,dn) is a repeating sequence of a space (X,d) which is not complete.

  1. If (Xn,dn) are compact metric spaces that converge to a compact metric space (X,d) in the Gromov–Hausdorff sense (this automatically implies that the spaces (Xn,dn) have uniformly bounded diameter), then the ultralimit is isometric to (X,d).
  2. Suppose that (Xn,dn) are proper metric spaces and that are base-points such that the pointed sequence (Xn,dn,pn) converges to a proper metric space (X,d) in the Gromov–Hausdorff sense. Then the ultralimit is isometric to (X,d).
  3. Let κ≤0 and let (Xn,dn) be a sequence of CAT(κ)-metric spaces. Then the ultralimit is also a CAT(κ)-space.
  4. Let (Xn,dn) be a sequence of CAT(κn)-metric spaces where Then the ultralimit is real tree.

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