Uniformly locally o-minimal structures of the second kind and their tame topology (New developments of transformation groups)

Abstract

This paper is a brief survey on a uniformly locally o-minimal structure of the second kind for non-specialists of model theory. An o-minimal structure enjoys tame topological properties such as monotonicity theorem and definable cell decomposition theorem. A uniformly locally o-minimal structure of the second kind is a new variant of o-minimal structure. A uniformly locally o-minimal structure of the second kind enjoys the local versions of tame topological properties possessed by an o-minimal structure. It enables to develop a tame dimension theory for definable sets.