Alexandrov spaces are CS sets

TL;DR

This paper proves that Alexandrov spaces possess a CS stratification structure and characterizes their spaces of directions, enhancing understanding of their geometric and topological properties.

Contribution

It establishes that the extremal stratification of Alexandrov spaces is a CS stratification and describes the topology of their spaces of directions.

Findings

Extremal stratification of Alexandrov spaces is a CS stratification.

Spaces of directions in certain Alexandrov spaces are homeomorphic to spheres.

In polyhedral cases, iterated spaces of directions also have spherical topology.

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.