Topology of the space of locally separable functions

Abstract

Among functions of n variables, the simplest are certainly those which are the sum of n functions, each of them depending only on one variable. Nevertheless, the set consisting of such particular functions is rather small compared to the whole space of general functions. Now, if we relax this condition and only ask for a local separability in some well chosen coordinates, is this request also binding and is the corresponding set of functions, that we will denote by S, small again? In this paper we will show that it is not actually the case and that S has a large set of interior points that we try to identify as well as possible.