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http://arxiv.org/abs/1207.3668 On Toponogov's comparison theorem for Alexandrov spaces Urs Lang, Viktor Schroeder (Submitted on 16 Jul 2012 (v1), last revised 25 Jul 2012 (this version, v2))
In this expository note, we present a transparent proof of Toponogov's theorem for Alexandrov spaces in the general case, not assuming local compactness of the underlying metric space. More precisely, we show that if M is a complete geodesic metric space such that the Alexandrov triangle comparisons for curvature greater than or equal to k are satisfied locally, then these comparisons also hold in the large. The proof is a modification of an argument due to Plaut.
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