Topology in nonlinear extensions of hypernumbers

Abstract

Modern theory of dynamical systems is mostly based on nonlinear differential equations and operations. At the same time, the theory of hypernumbers and extrafunctions, a novel approach in functional analysis, has been limited to linear systems. In this paper, nonlinear structures are introduced in spaces of real and complex hypernumbers by extending the concept of a hypernumber. In such a way, linear algebras of extended hypernumbers are built. A special topology of conical neighborhoods in these algebras is introduced and studied. It is proved that the space of all extended real hypernumbers is Hausdorff. This provides uniqueness for limits what is very important for analysis of dynamical systems. It is also proved that construction of extended real hypernumbers is defined by a definite invariance principle: the space of all extended real hypernumbers is the biggest Hausdorff factorization of the sequential extension of the space of all real numbers with the topology of conical neighborhoods. In addition, this topology turns the set of all bounded extended real hypernumbers into a topological algebra. Other topologies in spaces of extended hypernumbers are considered.