On the Geometry of sets satisfying the Sequence Selection Property

TL;DR

This paper investigates the geometric properties of sets satisfying the Sequence Selection Property (SSP), establishing transversality theorems and demonstrating the SSP's relevance in complex analytic and o-minimal structures.

Contribution

It introduces new transversality theorems and an SSP-structure preserving theorem, advancing understanding of the geometric and Lipschitz properties of SSP sets.

Findings

Transversality theorems for sets with SSP in singular cases

SSP-structure preservation under certain conditions

Applications to complex analytic varieties and o-minimal structures

Abstract

In this paper we study fundamental directional properties of sets under the assumption of condition (SSP) (introduced in a previous paper). We show several transversality theorems in the singular case and an (SSP)-structure preserving theorem. As an illustration, our transversality results are used to prove several facts concerning complex analytic varieties. The (SSP)-property is most suitable for understanding transversality in the Lipschitz category. This property is shared by a large class of sets, in particular by subanalytic sets or by definable sets in an o-minimal structure.