On the geometry of sets satisfying the sequence selection property

Abstract

In this paper we study fundamental directional properties of sets under the assumption of condition (SSP ) (introduced in [3]). We show several transversality theorems in the singular case and an (SSP )-structure preserving theorem. As a geometric illustration, our transversality results are used to prove several facts concerning complex analytic varieties in 3.3. Also, using our results on sets with condition (SSP), we give a classification of spirals in the appendix 5.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.2010 Mathematics Subject Classification. Primary 14P15, 32B20, Secondary 57R45.

(SSP) geometry with directional homeomorphisms

“…We have the following characterisation of condition (SSP ). As mentioned in [6], the proof in the relative case is similar to the non-relative case, for which we gave a detailed proof in [5].…”

“…Following the above works, we have started to work on condition (SSP ) both on the field of real numbers and on the field of complex numbers. In fact, we proved essential directional properties of sets satisfying (SSP ) with respect to bi-Lipschitz homeomorphisms in [6]. Amongst the main results in [6] are the following:…”

“…In fact, we proved essential directional properties of sets satisfying (SSP ) with respect to bi-Lipschitz homeomorphisms in [6]. Amongst the main results in [6] are the following:…”

“…Below we give several examples of sets satisfying the condition (SSP). Consult [6] for more examples.…”

“…Concerning 2, we proved two types of (SSP) structure preserving theorems in [6]. The main purpose in this paper is to introduce a new notion of homeomorphism, called directional homeomorphism, which enables us to show general results including those theorems mentioned above, using the new notion of homeomorphism without the assumption on the sequence selection property.…”