I am currently trying to understand "Curvature bounded below: a definition a la Berg--Nikolaev" by Nina Lebedeva and Anton Petrunin.
They start with a complete, intrinsic metric and space $X$ and say that one can assume that $X$ is geodesic (a shortest path exists between two arbitrary two points) otherwise one passes to the Ultraproduct of X. My question is what is the Ultraproduct of $X$ and why is it geodesic?
Wikipedia says that for metric spaces the Ultraproduct is somewhat more special and one considers the Ultralimit... I found something about Ultralimits in Metric spaces of Non-positive curvature in particular the following 3 results:
- Every Ultralimit of metric spaces is complete.
- The Ultralimit of a sequence of metric spaces is a length space if every metric space in the sequence was a length space.
- The Ultralimit of a sequence of metric spaces is a geodesic space if every metric space in the sequence was a geodesic space
However they do not explain, why I am allowed to consider a geodesic space if I start with just an intrinsic one.
Thanks in advance!