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:

    “Southerners, whose ancestors a hundred years ago knew the horrors of a homeland devastated by war, are particularly determined that war shall never come to us again. All Americans understand the basic lessons of history: that we need to be resolute and able to protect ourselves, to prevent threats and domination by others.”
    —Jimmy Carter (James Earl Carter, Jr.)

    “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)