On a Unified Analysis in the language of preordered sets

TL;DR

This paper introduces a unified framework using preordered sets that encompasses various fundamental concepts in analysis, providing a single theorem that explains properties like additivity and linearity across multiple notions.

Contribution

It presents a novel unified approach to analysis concepts through preordered sets, generalizing ideas from topology, group, and measure theories.

Findings

A single theorem explains additivity and linearity of multiple analysis notions.

The framework unifies concepts like limits, continuity, and integrals.

Generalization includes ideas from topology, group, and measure theory.

Abstract

We define a notion which contains numerous basic notions of Analysis as special cases, for example limit, continuity, differential, Riemann and Lebesgue integral, root and exponential functions. Properties like additivity or linearity of all mentioned notions follow from one single theorem. A generalisation of Definition 1 contains basic ideas of other theories such as topology, group and measure theory.