Arbitrarily Close for Summer 2022 Analysis
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.
Cite
@article{arxiv.2206.01181,
title = {Arbitrarily Close for Summer 2022 Analysis},
author = {John A. Rock},
journal= {arXiv preprint arXiv:2206.01181},
year = {2022}
}
Comments
This article overlaps considerably with the new version 2 of article 1912.13159 (Arbitrarily Close) which was submitted August 2022. Both articles have the same author and are on the same topic. Furthermore, article 2206.01181 has many typos and errors which have now been corrected in article 1912.13159