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October 8, 2019 / porton

Algebraic General Topology, a New Branch of Mathematics (Book) and How It Is Related With Limit of Discontinuous Function

mathematics

Algebraic General Topology (a book series for both postdoctorals and college students) is a new branch of mathematics that replaces General Topology. Yes, general topology is now legacy! We have something better than topological spaces, funcoids. You almost spent time in vain studying topological spaces: In not so far future colleges will teach funcoids courses instead of topological spaces courses, topological spaces will remain only in very specialized courses. Funcoids behave better than topological spaces.

The author of the book managed to formulate general topology (continuity, separability, convergence/limit, connectedness, total boundness, etc.) in terms of algebraic formulas. It is done without using topological spaces (however, topological spaces are also considered in the book) using the author’s concept of funcoids and reloids.

Funcoids and reloids replace and generalize all of the following:

  • topological spaces
  • pretopological spaces
  • proximity spaces
  • uniform spaces
  • even (directed) graphs

Yes, properties of topologies and graphs are described by the same formulas! We have a common generalization of topology/calculus and discrete mathematics.

For example, this is a formula (in the book there are three!) of all kinds (“regular”, uniform, proximal, discrete, others) continuity: fabf. Here f may be a function and a and b are spaces (be it topological, uniform, etc.) between which the function acts. In fact, continuity is defined for every partially ordered (pre)category.

For a further surprise, there is a formula for (generalized) limit of arbitrary (discontinuous) function:

lim f = { ν ∘ fr | rG }.

Now in the book’s system every function (between a wide class of spaces) is differentiable, every integral can be taken. And these limit do behave well: For example, if y is a value of the generalized limit, then yy = 0. Just open your mouth and try to realize the revolution that expects calculus soon.

Before going to the author’s discoveries, the book teaches the basic order, category, group theory and some of the legacy general topology, in order to be readable by anyone who knows basic set theory (and basic calculus to understand what the book is about).

After this the book presents some minor new results on order theory.

Then the author goes to the topic of filters on sets, lattices, and partially ordered sets. No doubt, this book is the world best reference on the topic of filters. Moreover, the author does not stop on the topic of filters on posets, but considers their generalization, filtrators. Filtrators is a very simple (it is just a pair of a poset and its subset) but powerful concept: most of the properties of filters do generalize for filtrators. The book reveals many previously unknown things about filters.

Then it starts the most interesting thing in the book: the theory of funcoids. Starting with an informal introduction, the author then considers funcoids in deep. It appears that funcoids are simultaneously a generalization of topological spaces, pretopological spaces, proximity spaces, (directed) graphs. The usual theory of topological spaces included but in the more general form of funcoids instead of spaces.

Then it follows the theory of reloids. Reloids is a very simple thing: a reloid is a filter on a Cartesian product of two sets. This is a generalization of the well known concept of uniform space. As you may know, uniform spaces describe such things as uniform continuity and total boundness on metric spaces.

The most interesting thing with funcoids and reloids is that they form a kind of algebra. So the name algebraic general topology. I have already shown you the formula (I remind: one of three formulas) of continuity. The author says that his main intention was to clean the mess: general topology was a mess of formulas with quantifiers where everybody could be lost, now it is instead an algebra, a beautiful theory.

The next chapter of the book considers interrelations between funcoids and reloids. And there are surprises.

The book considers even pointfree topology (topology without “points” or “numbers”). But not frames and locales, but pointfree funcoids instead, a simple easy generalization of funcoids.

The next thing in the book is kinda “multidimensional” general topology. The traditional point-set topology was kinda 2-dimensional, the author considers the arbitrary infinite dimensional topology (and yes, it does have applications, as applying multiple argument functions to limits of discontinuous functions needs this knowledge). And this infinite dimensional topology is also pointfree in the book.

Finally, the author presents a fascinating life story of the discovery. The main formula was discovered on the streets by a hungry homeless… This simple formula could have been discovered in 1937 but nobody except of a homeless first-year college student (the author) was able to guess it. Funcoids can be defined by four axioms that are easier than axioms of group theory, and nobody was able to guess.

Buy the book now (link in the beginning of this blog post), to jump to the very frontiers of the math research, whether you are a postdoctoral or a first-year student of a college.

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

Donate for this math research!

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September 20, 2019 / porton

Limit of Discontinuous Function

No root of -1? No limit of discontinuous function?

Like as once roots were generalized for negative numbers, I succeeded to generalize limits for arbitrary discontinuous functions.

The formula of limit of discontinuous function is based on algebraic general topology, my generalization of general topology in an algebraic way.

The formula that defines limit of discontinuous function is surprisingly simple:

lim f = { ν ∘ fr | rG }.

(This formula can be enhanced in different ways to make it behave better algebraically, but the idea is this.) And yes, it is very good algebraically, for example yy = 0 as if it were just a real number!

Read my book Algebraic General Topology.

So we can for example, define derivative of an arbitrary real function. It opens a way to wholly new discontinuous calculus.

There is a problem however: I didn’t yet succeeded to put this generalized derivative into a differential equation, because the left and the right parts of the equation would belong to different sets and we could not simply equate them. In my opinion, it is probably the most important current problem in all mathematics to invent a way to put my derivative into a differential equation.

What will happen next? I don’t know, but maybe for example, we will discover what is the structure at the point of singularity in a black hole.

September 10, 2019 / porton

The book “Algebraic General Topology” published officially

After a long time of being an unaccepted genius, the first volume of my book Algebraic General Topology is published officially (by the biggest Russian science publisher INFRA-M).

The most general in general topology and algebraic theory, generalization of limit for multivalued discontinuous functions, algebraic formula of continuity (for multivalued functions), common theory of calculus and discrete math + some other discoveries.

Here is the 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.

It is the first edition, wait for more to come.

Here is the link to the published book.

July 30, 2019 / porton

Monograph “Algebraic General Topology. Volume 1” sent for publication

I’ve sent the final version of the first edition of my research monograph Algebraic General Topology. Volume 1 (download here) to Russian publisher INFRA-M and signed the publication contract. They are going to publish my book electronically. They also asked to send them a Russian translation of my book to publish both in print and electronically.

The monograph contains the biggest discovery in general topology since 1937 (when filters were discovered).

Here is the abstract of the book:

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.

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.

Here is the fascinating story of my life while it was discovered.

Now the discovery is finally sent for an official publication after many years.

July 20, 2019 / porton

Algebraic General Topology. Volume 1 updated

The online book Algebraic General Topology. Volume 1 was updated.

The update corrects a few errors in the section “Equivalent filters and rebase of filters” (which was previously marked as unchecked for errors).

July 9, 2019 / porton

The organization EULER Foundation is discriminatory

The organization EULER Foundation and personally their president Vostokov Sergey Vladimirovich are discriminatory. Moreover as they accept donations from general public, they are cheaters because they don’t warn the donors that their money is going to be used discriminatory.

They say on their site that they help to mathematicians and their family members in need.

Accordingly, I’ve sent them the following letter (as translated from Russian):

Hello!

It’s written on your site that you are helping the needy mathematicians.

This is exactly the case. I need money for a lawyer. The claim is attached.

The remaining documents will send on request

To the email there was attached my court appeal (here it’s its English translation).

Their reply:

Forget this address,

So, they have a discriminatory policy toward Christians, as they read in my attached court appeal that I am a Christian.

June 24, 2019 / porton

I am a genius that is all the rest are idiots (an article with too simple math formulas)

I made several scientific discoveries (mainly in abstract mathematics but also in XML files processing). I claim to be a genius based on this fact. But what I mean saying “I am a genius”? I mean just that all the rest experts showed themselves… idiots. Most of the formulas are too simple. Nobody guessed, nobody followed the rules taught to sophomores if not freshmen.

The first thing I discovered is the theory of filters. Mathematicians defined filters on sets and more generally on lattices and on posets and proved some theorems about them. Many tens of years nobody wrote an article with a systematic study of filters. I did this. I did this before I knew how filters are related with topologies (Stone duality).

Later I discovered what I call “filtrators”. A filtrator is just a pair of a poset and its subset (yes, that’s the definition of a filtrator, can it be easier?) If we take the poset the set of filters ordered reversely to set theoretic inclusion and the subset the set of principal filters, then essentially the set of filters becomes a filtrator. So I generalized many of my theorems about filters for filtrators.

My main math discovery is “funcoids”. I did it at the first year of a university. I decided: I will discover an algebraic way of general topology. I tried to understand the “mystery” of topological spaces. A topological space maps points to filters. I generalized it to mapping filters to filters and wrote the following formulas (note that I changed the notation below to be easier to explain instead of my first year self-contradictory notation):

  • \Delta\varnothing = \varnothing (I equated sets such as \varnothing with principal filters.)
  • \Delta(A\sqcup B) = \Delta A\sqcup \Delta B
  • \Delta A\sqsupseteq A (or was it initially \Delta\{x\}\sqsupseteq\{x\}?)

Here \sqcup, \sqcap, \sqsupseteq are lattice and poset operators on the poset of filters. A and B denote arbitary filters.

Can it be simpler? A sophomore could write these formulas.

The third formula was superfluous and I later removed it.

The next step was to consider two such “Deltas”: \Delta_1 and \Delta_2 and realize they bijectively correspond to each other by the formula: \Delta_1 A\sqcap B\ne\varnothing \Leftrightarrow \Delta_2 B\sqcap A\ne\varnothing.

It was almost the definition of a funcoid. To make it (almost) the modern definition of funcoid, it is enough to take \Delta_1 and \Delta_2 arbitrary functions on the set of filters on some fixed set. Then the formulas \Delta\varnothing = \varnothing and \Delta(A\sqcup B) = \Delta A\sqcup \Delta B follow automatically.

I can be proud that I did this before I became a sophomore.

Then I proved that a funcoid can be alternatively (up to an isomorphism, if I knew the word isomorphism that time, I am not sure) defined as a binary relation \delta between sets conforming to the axioms:

  • \lnot(\varnothing \delta B)
  • \lnot(A \delta \varnothing)
  • A\cup B \delta C \Leftrightarrow A\delta C\vee B\delta C
  • C \delta A\cup B \Leftrightarrow C\delta A\vee C\delta B

The above formulas are worth the title of a genius. Many tens of years nobody guessed to write them down. Every third year student would be able to.

Then I developed the theory of funcoids in details. It was an extremely interesting journey with many personal discoveries. Some conjectures are yet unsolved.

Another “big” thing I guessed was to consider the set of filters on a cartesian square of a set. I proudly called this “reloids” (as a generalization of binary relations). It seems nobody before me researched this obvious in idea topic. Reloids are comparably interesting to funcoids (however, I’d say funcoids are my main discovery). I tried to research reloids like funcoids in as much details as I can.

Then I discovered than funcoids and reloids are related and founds several mappings between the class of funcoids and the class of reloids. First I mapped reloids into funcoids with a certain maps that I denoted (\mathsf{FCD}). It was a stunning for me discovery that there are two distinct “easy and natural” ways to map funcoids into reloids. I also researched all these three mappings and their relations with each other.

There were many interesting details and specific “discoveries” on the way, but the above describes the essence of this most starting stage of my research. I will also note that I soon discovered that both funcoids and reloids are partially ordered (and are even lattices).

Pointfree funcoids were a simple but not quite trivial for me to guess generalization of funcoids. For them I defined a partial order in a specific case by a trivial generalization of the order on the set of funcoids. Later I generalized this order for arbitrary pointfree funcoids with a simple but taking time to guess formula.

One more big idea was “staroids”. Staroids are are generalization of pointfree funcoids. Definition is also simple but hard to guess and took some time to write down in the modern concise form. Later I also guessed prestaroids and completary staroids. Another man proved that completary staroids are not the same as staroids. Another because I am bad at solving problems and particularly at finding counter-examples, what he did.

Before staroids I also discovered related concept, “multifuncoids”.

It took some time to research some properties of staroids in many details. But in great degree staroids are a mystery yet.

Yet later I discovered the remarkable fact that the set of funcoids is isomorphic to the set of filters on a certain (boolean) lattice of sets. Am I the first person who described this simple lattice in details?

I also discovered continuity and connectedness for funcoids and for reloids. Continuity is remarkable by being described by simple algebraic (without quantifiers and epsilon-delta) formulas. It generalizes continuity, uniform continuity, and proximity continuity in the same formulas. I was greatly excited by this discovery. I found that these three formulas coincide for monovalued functions (and monovalued funcoids/reloids, which I later also described in simple algebraic way) but continuity of non-monovalued functions/funcoids/reloids is defined by me in three different and non-equivalent algebraic formulas. Yes, I defined continuity for non-monovalued functions.

Connectedness definitions also provided several yet unsolved open problems.

I easily defined limit of a funcoid, generalizing limit of a function on a filter.

The next stunning thing was generalized limit. I defined limit for an arbitrary (discontinuous) function and even for an arbitrary funcoid (as such it includes also any multivalued function). Here is the extermely simple (but not easy to guess) formula:

\lim f = \{ \nu\circ f\circ\uparrow r \mid r\in G \}.

Here \nu is a topological structure (funcoid) on the space where our function or more generally funcoid f acts to and G is a group such as the group of translations of our “source” vector topological space or like this.

Not too hard to prove that for continuous functions the generalized limit and customary limit are bijectively related. The generalized limit has algebraic operations (like addition, multiplication and like) which easily extend the operations on the points of our space.

Now we have every function differentiable (just like as we have square root of -1). But I didn’t succeed (yet) to put this generalized derivative into a differential equation, because the set of values of the limit is not the same as the set of values of the differentiated function. I tried this but it appeared hard: a function with singularity when differentiated produces an “infinitely big” derivative. If we differentiate this derivative, it becomes an “infinity of infinities”, etc. We have a countable sequence of bigger and bigger kinds of infinities or singularities (name it whatever you want). In my opinion, it is probably the most important problem of modern mathematics how to put them into both left and right part of a differential equation and define what is equality for them. I have only guesses what are non-smooth and discontinuous differential equations. Start working on this problem!

I wrote a book about all this. I wrote also some draft notes for the future and a partial draft of the second volume of the book. The second volume contains category theory things related with my theory. In the way I despite not being an expert in category theory discovered some new fundamental category theory concepts.

Also I introduces some simple but novel order theory concepts and theorems.

I wrote all above into an 393 pages book (also contains introductory materials for beginning students) + addons and second volume. I omitted many particular details above. It is freely available at my site.

Later I discovered a way to turn a category (with certain additional structure) into a semigroup. This way I “translate”/generalize funcoids and reloids from the language of category theory into the language of semigroup theory. In the way I understood what are “identities” in a rather wide generalization of them.

I also discovered two things unrelated to this:

  1. Theory of formulas. It is a very simple axiomatic system describing anything with different kinds (like the left and the right part of a formula) with constituent parts (like subformulas). I only started this research. I strongly suspect that I discovered something related with trinity of God. It has three parts and these parts seems to describe everything and are somehow related with things like logic and thus reason. Somethings like three ways to reason about anything or like this.
  2. A new way to process XML files. I dreamed for a long time to make processing of XML files in several stages based on elements namespaces automatically. Implementing this nearly obvious idea (I think I am not the first person thinking of it) turned out to be a work on many details with no big subideas, but many small but interesting subideas. I wrote a relatively complex data and algorithm specification. This is dissimilar to my math research where I had big ideas, but small ideas were less important. I was able to do both kinds of things: big ideas in math but details in programming. I successfully implemented the most important parts of in Python programming language. My command line utility took about a second to start what is too much. I tried several programming languages and finally chose D language to rewrite it in it to make it fast (and more reliable). This is a work in progress. It will replace HTML and other modern XML technologies. Yes, that’s the future, a specific way to implement the dreams of people who invented XML for it.

I also made many theological discoveries in Bible study, including a new religion and a new technological way to study Old Testament.

Oh, in the way once I acted like a crackpot. I assumed that I made an even greater discovery that I called “21st century math method” which I put without a proof at my site. Later I found these unproved theorems erroneous and deleted it from the site.

The conclusion: I made a series of discoveries worth at least trillions of dollars in this way. But if it would come that nobody guessed yet 50 years, it may cost more. I am probably two times more smart than Einstein. Only I guessed easy formulas, only I followed the sophomore rules. Special relativity would be guessed by somebody other in a few years. But to repeat my discoveries it takes tens of years. I closed a big hole in the humankind’s mathematics.

Now I want to win the legal battle with Russia.