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:

    “When you realize how hard it is to know the truth about yourself, you understand that even the most exhaustive and well-meaning autobiography, determined to tell the truth, represents, at best, a guess. There have been times in my life when I felt incredibly happy. Life was full. I seemed productive. Then I thought,”Am I really happy or am I merely masking a deep depression with frantic activity?” If I don’t know such basic things about myself, who does?”
    —Phyllis Rose (b. 1942)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)