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PAPERBACK // E-BOOK // PDF (free)

This research was peer reviewed and published by INFRA-M.

See also Book 3: Algebra; Book 4: Limit of a Discontinuous Function.

Algebraic General Topology—A Generalization of Traditional Point-Set Topology

Algebraic General Topology (AGT) is a wide generalization of general topology, allowing students to express abstract topological objects with algebraic operations. The book consists of definitions, theorems and proofs of this new field of math.

This author attempts to discover new unexpected results in point-set topology with AGT, thus, revolutionizing the concepts of limit of discontinuous function, integration and differential equations. This not only simplifies existing research method, but also consequently radically changes the fields of natural science, such as engineering, physics and economics.

This new research field generalizes new theorems (which do not have analogs) as well as former analysis by collapsing several theorems of analysis into one AGT equation. The book presents a new take on the classic theories of uniform spaces and topological spaces.

I’ve discovered Algebraic General Topology ( AGT ), a new field of math which generalizes old General Topology. Mathematical Synthesis is how I call* Algebraic General Topology applied to study of Mathematical Analysis.

Algebraic General Topology. Volume 1 (Paperback book) (published by INFRA-M, updated). My theory as a book, starting with basic math, so even novices can read. You are strongly recommended to read this book, not “legacy articles” below. It is a studybook on general topology but with my own more general theory instead of the classic theories of topological spaces and uniform spaces.

PDF Slides [HTML] prepared by me in order to give a talk at a research conference. Use these slides to quickly familiarize yourself with basics of Algebraic General Topology. (Please speak instead of me at some math conference, because my spoken English is bad and I yet have no money to travel. Just contact me.) and even shorter introduction [HTML] (PDF).

This comprehensive multi-volume book deals with general topology, broadly conceived. This book offers an advanced course in general topology. It discusses topological properties such as limit points and continuity, Connectedness and Separation axioms.

Algebraic Topology provides the most general approach to the subject. With coverage relevant topics and their applications to theorems of point-set topology, this book is perfect for communicating complex topics and the fun nature of general topology for beginners.

This book aids the reader to gain very significant foundation in mathematics: Despite its success as a mathematical tool, in the general theory of topology and its adaptability to a wide range of mathematical and physical problems. This book is an attempt to broaden this point of view at the stage where the student first encounters the subject.

This book is suitable for the reader without a deep mathematical background. Simple Approach was considered by the authors as a continuation of advanced calculus, thus achieving the right balance between formal and abstract approaches to the topic.

This unique book on General Topology addresses practical issues of uniform spaces, metric spaces, topological spaces and many more. The book covers newly-developed fields like General topology (point-set and pointfree topology).

See also:

Project description

Overview

Algebraic General Topology (= theory of funcoids, reloids, and their generalizations) is a wide generalization of general topology, allowing to express topological properties by algebraic formulas.

Victor Porton wrote his book “Algebraic General Topology. Volume 1” containing a description of this theory (with definitions, theorems, and proofs, as well as expository material allowing its reading by beginning students).

Intellectual Merit

Algebraic general topology is obviously an exceeding breakthrough because:

It is a new big branch of mathematics.

It is a very general theory based on humorously simple axioms.

It is expressed in algebraic form, not just a mess of quantifiers as old general topology.

It contains among other a definition of (generalized) limit of discontinuous function, what is expected to make a revolution in such fields as differential equations.

The general topology was generalized for an arbitrary kinda multidimensional case (while traditional general topology appeared to be 2-dimensional).

The idea of turning a category with additional structure to a semigroup is also novel.

The definition of funcoid is the biggest discovery in general topology since 1937 (when filters were defined).

The work also contains a generalization of filters on posets and other significant discoveries.

It is a common generalization of calculus and discrete mathematics.

Common generalization of continuity, uniform continuity, proximal continuity, Cauchy-continuity, discrete continuity.

It is connecting two previously thought unrelated branches of mathematics: general topology and ordered semigroup/monoid theory. Ordered semigroups are used to describe general topological properties in a novel way; general topology and calculus bring new concepts (such as three kinds of continuity or monovaludness) into ordered semigroup theory.

Author proposed a new way to define “sides” of manifolds and related things in a more natural way. We previously used the word “side” of a surface but didn’t really know what it really is.

The book is a good studybook. Also it is the world best reference on the topic of filters on posets.

Broader Impacts

As author’s experience shows that we discover new unexpected results in traditional point-set topology using author’s method. This tendency is expected to continue. General topology and thus related branches of science change radically.

(Generalized) limit of discontinuous function is expected to revolutionize differential equations, integrating, etc. and consequently such natural sciences as physics, engineering, and economics. It also simplifies existing research methods.

The project implies broad dissemination of my knowledge to the entire mathematical community to foster further science development.

The resulting book can serve as a base of college courses on different topics of algebraic general topology and for a college course on filters on posets and lattices, because it is written as a studybook.

If you are a teacher, you can make the following college courses using it as a studybook:

basic order theory

(co-)brouwerian lattices

filters and filtrators

funcoids

reloids

interrelationships between funcoids and reloids

multidimensional general topology

and more

Book abstract

In this work I introduce and study in details the concepts of funcoids which generalize proximity spaces and reloids which generalize uniform spaces, and generalizations thereof. The concept of funcoid is generalized concept of proximity, the concept of reloid is cleared from superfluous details (generalized) concept of uniformity.

Also funcoids and reloids are generalizations of binary relations whose domains and ranges are filters (instead of sets). Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this provides us with a common generalization of calculus and discrete mathematics.

It is defined a generalization of limit for arbitrary (including discontinuous and multivalued) functions, what allows to define for example derivative of an arbitrary real function.

The concept of continuity is defined by an algebraic formula (instead of old messy epsilon-delta notation) for arbitrary morphisms (including funcoids and reloids) of a partially ordered category. In one formula continuity, proximity continuity, and uniform continuity are generalized.

Also I define connectedness for funcoids and reloids.

Then I consider generalizations of funcoids: pointfree funcoids and generalization of pointfree funcoids: staroids and multifuncoids. Also I define several kinds of products of funcoids and other morphisms.

Before going to topology, this book studies properties of co-brouwerian lattices and filters.

Open problems, research topics, and related stuff

If you solve any of these problems, please notify me! Also (half jokingly) read here

Research in the middle section at Virtual scientific conference site.

Singularities

a Wiki dump about singularities [HTML]

Reduced limits [HTML] (very rough partial draft) – Operations with values of singularities. Conjectures about application of this to General Relativity and black holes.

Cartesian closed categories

Wiki about proving certain categories are cartesian closed

Certain categories are cartesian closed [HTML] (rough draft with errors)

Other

Short note about limit of discontinuous function – generalization of limit for noncontinuous functions

Open Problems in AGT [HTML] (PDF) – This document lists all yet unsolved problems and conjectures in the field of AGT .

Some of my further research directions (on PlanetMath)

Partial proofs [HTML] (PDF, very, very rough draft) – Partial proofs (with rough gibberish) about open problems I have tried to solve but have failed.

Funcoids and Reloids project at the projects site

On a common generalization of funcoids and reloids

Correct errors in Compact funcoids [HTML].

Misc

Todd Trimble’s commentary/notes on my work (an alternative view on my theory) (see also my response [HTML] on Todd’s notes)

Errata for my published works [HTML]

Quasi-cartesian functions [HTML] – a blind valley of research, a theory which I am not going to publish.

Grant request sent to fpi.ru fund [HTML] (in Russian) – a good description why my work is revolutionary.

A short explanation what Algebraic General Topology and Math Synthesis are

Algebraic General Topology is about how to act with abstract topological objects expressing infinities with algebraic operations.

AGT is to math/functional analysis like algebra to old prose arithmetic.

Achievements and advantages:

general topology expressed in simple algebraic operations

simplicity of operating with infinities, as infinities now can be comprehended as something “whole”, not a mess of parts

two-three line proofs of some old pages length analysis theorems

multivalued functions are now so simple to study as single valued

frees analysis from its messy epsilon-delta notation

analysis of non-continuous functions

partially formally unifies math analysis and discrete mathematics

not limited in any way to metrizable spaces and countable sets

generalized limit of discontinuous function

We now we can get rid of math analysis as now it becomes synthesis, I would say. So I call AGT applied to study of such things as continuity, limits, and differentials Mathematical Synthesis.

AGT isn’t a continuation of former functional analysis research, it is re-research anew beginning almost from scratch. This makes the knowledge level of the First course of a math faculty enough to understand it.

This new research field is both just generalizing former analysis and new theorems/concepts not having analogs in old theories. Several different theorems of analysis often collapse into one AGT equation of which they are obvious consequences.

AGT is very abstract, indeed even the current level of AGT knowledge often allowed me to find simple solutions of practical tasks (such as calculations of infinite sums). I have not yet reached the level of integrals in the synthesis research.

AGT is a kind of thinking with formulas. No real numbers analysis expressiveness with visual images preserved. That is not needed anyway as the equations of AGT are even more clear than graphics of old analysis. AGT is simple, natural, and beautiful.

Note that Algebraic General Topology being a generalization of General Topology has nothing in common (except of the name) with Algebraic Topology. Math synthesis is a generalization of functional analysis.

Misc

Funcoids at nLab.

Algebraic General Topology at Facebook.

Algebraic General Topology at WikInfo.

Outdated materials

Use the above materials instead, this section is outdated.

New research (not included in the book), most of it is also available as the above PDF files

Legacy AGT articles (read the book instead)

* Hm, probably I should not call it so. The term mathematical synthesis was sometimes used for analysis with Fourier analysis.

To know more about Algebraic general Topology, visit:

https://mathematics21.org/2019/12/23/what-is-algebraic-general-topology-video/

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