The ultralimit with respect to ω of the sequence (Xn,dn, pn) is a metric space defined as follows.
As a set, we have .
For two -equivalence classes of admissible sequences and we have
It is not hard to see that is well-defined and that it is a metric on the set .
Denote .
Read more about Ultralimit: On Basepoints in The Case of Uniformly Bounded Spaces, Basic Properties of Ultralimits, Asymptotic Cones, Examples