*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).

See also:

- Addons which should be added to volume 1 [HTML] (a very rough partial draft)
- Volume 2 [HTML] (a very rough partial draft)
- Research ideas [HTML] (a list of short ideas)
- The Story of the Discovery [HTML] (maddest story ever)

## 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

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

**Products in dagger categories with complete ordered Mor-sets [HTML]**(draft)- I define a categorical direct product in the category of continuous maps between endofuncoids and some other similar categories.
**Equalizers and co-Equalizers in Certain Categories [HTML]**(rough partial draft)- I define a equalizers and co-equalizers in certain categories.
**Compact funcoids [HTML]**(PDF, preliminary partial draft with errors)- Compactness of funcoids generalizing compactness of topological spaces is defined.
**Pointfree funcoids as a generalization of frames [HTML]**(PDF, rough partial draft)- I define an order embedding from the set of frames to the set of pointfree endo-funcoids.
**Connectors and generalized connectedness [HTML]**(PDF, preprint, may be not a big discovery)- It’s defined the notion of connectedness for special binary relations called connectors. This generalizes topological connectedness, path connectedness, connectedness of digraphs, proximal connectedness, and some other kinds of connectedness.
**Categories related with funcoids [HTML]**(PDF, preliminary partial draft)- I consider some categories related with pointfree funcoids.
**Cauchy Filters on Reloids [HTML]**(PDF, preliminary partial draft)- I consider Cauchy filters on reloids, generalizing Cauchy filters on uniform spaces. Using Cauchy filters, I define Cauchy-complete reloids, generalizing complete uniform spaces.
**Hyperfuncoids [HTML]**(PDF, preliminary partial draft)- Staroids on powersets described as filters on certain lattices.
**Backward Funcoids [HTML]**(PDF, preliminary partial draft)- A new idea about staroids. A very preliminary draft.

### Legacy AGT articles (read the book instead)

**Dual filters and ideals [HTML]**- Sets of filters, of ideals, and yet two kinds of objects are isomorphic.
**Funcoids are co-frame [HTML]**(PDF, rough preliminary draft)- A proof that funcoids form a co-frame (without axiom of choices).
**Funcoids are filters [HTML]**(a partial draft)- Funcoids bijectively correspond to filters on a certain lattice.
**Another version of cross-composition of funcoids [HTML]**(PDF, draft)- Introduced another version of cross-composition of funcoids. This forms a category with star-morphisms. It is conjectured that this category is quasi-invertible.
**Funcoids and Reloids [HTML]**(PDF, preprint)- Let’s consider generalizations of proximity spaces and uniform spaces.
Also in this article continuity is defined in algebraic manner which hides old epsilon-delta notion under a smart algebra. It generalizes continuity, uniform continuity, and proximity-continuity in one formula.

**Orderings of filters in terms of reloids [HTML]**(PDF, draft)- Orderings of filters which extend Rudin-Keisler preorder of ultrafilters are defined in terms of reloids. Also there is defined isomorphism of filters which extends Rudin-Keisler equivalence of ultrafilters.
**Decomposition of composition [HTML]**(PDF, rough draft)- I prove that composition with a principal reloid is distributive over join of reloids. I do it using
decomposition

of composition of reloids into two operations.Then I research certain embedding of reloids into funcoids, in order to prove some properties of reloids.

**Identity Staroids [HTML]**(draft)- The concept of
*n*-ary identity relation is generalized to what I call identity staroids and identity multifuncoids. It is proved that staroidal product of ultrafilters can be non-atomic. I am going to integrate materials from this article into my book before its publication. **Convergence of funcoids [HTML]**(PDF, partial draft)- It’s defined the notion of convergence and limit for funcoids.
It’s defined (generalized) limit of arbitrary (not necessarily continuous) functions under certain conditions.

**Pointfree funcoids [HTML]**(PDF, draft)- Consider a generalization of funcoids.
**Multifuncoids [HTML]**(PDF, rough preliminary draft)- They are defined multifuncoids. Their basis properties are researched.
I define product of two morphisms for certain categories. (These products are pointfree funcoids.) As a special case of this product of two funcoids and product of two reloids is defined.

I haven’t yet shown that these are direct products in categorical sense.

As a generalization of product of two funcoids I define product of an arbitrary family of funcoids.

**Conjecture: Upgrading a multifuncoid [HTML]**(PDF, draft)- An additional open problem, said concisely: upgrading a multifuncoid is a multifuncoid. (The concepts of
*upgrading*and*multifuncoid*are defined in this article.)In this article I prove the conjecture for n=0,1,2. n=3 and above is unknown.

**Totally bounded reloids [HTML]**- A generalization of totally bounded uniform spaces. It splits into several distinct concepts which are equivalent for the special case of uniform spaces.
**Filters on Posets and Generalizations [HTML]**(a preprint incorporating errata). Published in IJPAM: open access PDF- It’s about filters on arbitrary lattices and posets as well as certain generalizations thereof.
**A note on starrish posets [HTML]**(preprint)- Some easy strengthening of some theorems from the article
Filters on Posets and Generalizations

. **Pseudodifference on atomistic co-brouwerian lattices [HTML]**(preprint)- Solved
problem 1

from Filters on Posets and Generalizations [HTML]. **Free Stars [HTML]**(preprint)- In this short article I investigate properties of free stars, first introduced in this article [HTML].
**A New Kind of Product of Ordinal Number of Relations having Ordinal Numbers of Arguments [HTML]**(preprint)- Infinite associativity is defined for functions taking an ordinal numbers of arguments. As an important example of an I define
*ordinated product*and research it’s properties. Ordinated product is an infinitely associative function.

* 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/