Alexandrov spaces are CS sets

Abstract

We prove that the extremal stratification of an Alexandrov space introduced by Perelman-Petrunin is a CS stratification in the sense of Siebenmann. We also show that every space of directions of an Alexandrov space without proper extremal subsets is homeomorphic to a sphere. In the polyhedral case, the same holds for every iterated space of directions.

revised incorrect terminology on PL structure in Theorem 1.4 and Lemma 5.1 and added Remarks 1.5 and 1.7