On Basepoints in The Case of Uniformly Bounded Spaces
Suppose that (Xn,dn) is a sequence of metric spaces of uniformly bounded diameter, that is, there exists a real number C>0 such that diam(Xn)≤C for every . Then for any choice pn of base-points in Xn every sequence is admissible. Therefore in this situation the choice of base-points does not have to be specified when defining an ultralimit, and the ultralimit depends only on (Xn,dn) and on ω but does not depend on the choice of a base-point sequence . In this case one writes .
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