Arbitrarily Close for Summer 2022 Analysis

TL;DR

This paper explores the concept of 'arbitrarily close' in analysis, providing foundational insights into limits, convergence, and topology, with applications to Euclidean spaces and continuous functions.

Contribution

It introduces a topological perspective on 'arbitrarily close' and applies it to fundamental concepts in calculus and real analysis, offering a new approach to classical results.

Findings

Provides a new foundational approach to limits and convergence.

Explores the topology of Euclidean spaces via closed sets.

Introduces basic properties of continuous functions in Euclidean spaces.

Abstract

The kernel of analysis, to me anyway, is the following idea: A point is arbitrarily close to a set if every neighborhood of the point intersects the set. Defining ``arbitrarily close'' in this way provides a foundation for classical results in calculus and real analysis dealing with convergence, limits, connectedness, limits, continuity, differentiation, integration, series, and more. This book contains: a thorough introduction to arbitrarily close; an approach to limits and convergence of sequences using arbitrarily close as a key first step; the topology of Euclidean spaces stemming from closed sets; an exploration of the properties of functions like domains, ranges, and images of sets and sequences; and an introduction to basic aspects of continuous functions between Euclidean spaces. Even so, ``arbitrarily close'' reaches deeper than discussed here. The idea is topological in nature and there is much more to explore.