External borders and strongly open sets

Abstract

External sets, in particular neutrices, were proposed as nonstandard mathematical models of vagueness. We introduce sets with external borders, characterized by the fact that the time spent within the set by paths leaving them ("border-crossings") is necessarily an external set. We show that sets with external borders are open. Sets are called strongly open if they are stable by balls of some fixed non-zero radius. In standard d-dimensional space they are stable by translation over a d-decomposable neutrix, i.e. a direct sum of d neutrices of R. Such sets clearly have external borders. A converse holds for precompact external sets with lowest complexity (unions or intersections of a family of internal sets indexed by the standard elements of a standard set).