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.

Read more about this topic:  Ultralimit

Famous quotes containing the words basic and/or properties:

    “The research on gender and morality shows that women and men looked at the world through very different moral frameworks. Men tend to think in terms of “justice” or absolute “right and wrong,” while women define morality through the filter of how relationships will be affected. Given these basic differences, why would men and women suddenly agree about disciplining children?”
    —Ron Taffel (20th century)

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)