# Algebraic General Topology and Math Synthesis

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.

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).

Algebraic General Topology. Volume 1 [HTML] (PDF, good draft, checked for errors). 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.

The LaTeX source of the book is available at a Git hosting. This means that you can clone my LaTeX files and help me to hunt errors, add new definitions and theorems, generalize existing theorems. The e-book is available free of charge, forever. The book will probably never be published "officially". If you want to cite my book, refer to a version of LaTeX files in the above Git repository.

## 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] (PDF, 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

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.

• 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

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 Google+.

## 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.