I’ve published a new edition of my book Algebraic General Topology. The new edition features “unfixed morphisms” a way to turn a category into a semigroup. (Certain additional structure on the category is needed.)

The book features a wide generalization of general topology done in an algebraic way. Now we can operate on general topological objects with algebraic operations.

The book features the concept of *funcoid* – something better than topological space. Topological spaces mostly become history, the new thing of general topology is funcoids. It also considers filters on the Cartesian product of sets, pointfree topology, “multidimensional” general topology, where traditional topological spaces are always two-dimensional in a sense, as they consider binary relations (a relation between one point and one set).

Completely overthrow your understanding of general topology with this book. The series of the book also presents even more abstract semigroup algebra of general topology and a generalization of limit for an arbitrary function.

]]>A discontinuity is any point on a function where one of the three possibilities arise:

- The right-side limit is unequal to left-side limit
- The function jumps suddenly
- The function goes to infinity at a certain point in the domain.

Continuity and limits, as understood in traditional calculus, rely on infinitesimally small sets to arrive at limits for arbitrary functions. This idea, when translated into other definitions of continuities, relies on ideas about convergence of sequences and the like.

Such an approach relegates analysis on discontinuous functions to the side-lines because these functions violate the principles of infinitesimal calculus and thus not worthy of consideration.

It’s understandable why mathematicians have given continuity and infinitesimally small sets this much import—calculations of a practical nature may seem much more relevant to our concerns. Even with these convictions, it doesn’t disqualify analysis of discontinuous functions from mathematical theorization.

One of the outcomes of Algebraic Generalized Topology is the generalization of limits of discontinuous functions. This is a relatively novel path, considering the extensive literature that relies on the fundamental theorem of calculus and the traditional understanding of continuity, but is no less relevant to mathematical analysis. It is a branch of mathematics that is capable of generalizing continuity as well as discontinuity across different mathematical spaces, effectively making mathematical analysis much more powerful than it is currently.

Funcoids and Reloids are the fundamental components of mathematical analysis in AGT. For the purposes of the reader, I think it would be prudent to go through the basic definitions. These include:

A relation from a set *A *to a set *B, *such that it is* *a quadruple (*A, B, α, β*) where *α *ϵ F(*B*)^{F}^{(}^{A}^{)},* α*ϵ F(*B*)^{F}^{(}^{A}^{) }such that:

∀*X*ϵF(*A*)*, Y*ϵF(*B*) : (Y∩*α*(X) ≠0 ⇔ *X*∩*β*(*Y*) ≠0)*.*

F(A) here denotes the set of filters on a set A.

A Reloid is defined as:

A Reloid from a set *A *to a set *B is *a triple (*A, B, F*) where,

F is a filter on the Cartesian product A x B of two sets

I intend to go into deeper explanation of my work in later publications, but, for perspective, I think it is also worthwhile to think of the definition of continuity in terms of Funcoids and Reloids. The definition is as follows:

* f ϵ *C(µ,ν) *f*** **∘µ ≤ ν∘ *f*

This definition applies to topological spaces and also to uniform spaces, proximity spaces, graphs, etc. I also take the concept of funcoids to create an elegant generalization of limits for arbitrary discontinuous functions.

I have also worked extensively with limits of discontinuous functions and mathematical nondifferentiable analysis in my math research monograph titled Algebraic General Topology. Anyone interested in my work should get in touch with me directly.

]]>The methods or syntax to express mathematical truths are, obviously, essential to the study of mathematics. Without a taxonomy that highlights the correct use of mathematical symbols, rules which dictate the validity of mathematical statements and theory about how these statements are to be construed, you can’t *do *math.

Within the philosophy of mathematics, there is much debate about the efficacy and the nature of our language to express certain truths, whether these truths imply ontological commitments and so on.

My Axiomatic Theory of Formulas furthers our understanding of mathematical logic and carries implications for the analysis of math formulas, objects in computer memory etc. Equipped with this knowledge, it’s entirely possible that we can create more advanced mathematical expressions.

I think the best way to describe the ATF is by speaking of it in the ways that I have in my publication titled “Axiomatic Theory of Formulas”. Above all else, my theory concerns mathematical formulas and specifically it describes a new way to speak of mathematical constructs.

In general terms, a construct is any statement with indexed components. It’s easier to understand mathematical constructs if I first explain how English sentences can be thought of as constructs:

“The Sky Is Blue”

The above statement has separate components constitute a proposition. It has indexed components in the following manner—“the” has index 1, “sky” has index 2, “is” has index 3 and “blue” has index 4. If we remove any of these components, either the statement will refer to a different proposition or it will become meaningless.

Similarly, in mathematics, the expression of the form *a*-(*b*+*c*) has components a, b and c. Here “a” has index 1, “-“ has index 2, while (b+c) is a sub-formula and the component with index 3. Additionally, a formula is a special case of mathematical constructs which contains symbols.

I specifically study the application of operators on sets of constructs, rather than the constructs themselves. Through the approach, I have derived some useful and interesting insights about the behaviors of these sets._{}

Anyone interested in the ways they can improve the performance of their computational algorithms or see how the ATF improves mathematical logic should get in touch with me directly. Additionally, you should also take a look at my Axiomatic Theory of Formulas to further explore the implication of my theory for mathematics. I have also worked extensively with limit of discontinuous function and mathematical non-differentiable analysis in my math research monograph titled Algebraic General Topology.

]]>**Abstract.** A review of my book “Generalized limit (of arbitrary discontinuous function)”. A popular introduction with graphs to the following topic:

I consider (a generalized) limit of an arbitrary (discontinuous) function, defined in terms of funcoids. The definition of the generalized limit makes it obvious to define such things as the derivative of an arbitrary function, integral of an arbitrary function, etc. My book also defines of non-differentiable solution of a (partial) differential equation. It’s raised the question of how do such solutions “look like” starting a possible big future research program.

**Keywords: **algebraic general topology, limit, funcoid, differential equations

**A.M.S. subject classification: **54J05, 54A05, 54D99, 54E05, 54E17, 54E99

In this informal article, I will explain popularly my concept of generalized limit of an arbitrary (discontinuous) function. For details and proofs see [1] and [2].

For example, consider some real function f from the *x*-axis to the *y*-axis:

Take it’s an infinitely small fragment (in our example, an infinitely small interval for x around zero; see the actual book for an explanation what is infinitely small):

Next consider that with a value y replaced with an infinitely small interval like [*y* – ϵ; *y* + ϵ]:

Now we have “an infinitely thin and short strip”. In fact, it is the same as an “infinitely small rectangle” (Why? So infinitely small behave, it can be counter-intuitive, but if we consider the above meditations formally, we could get this result):

This infinitely small rectangle’s y position uniquely characterizes the limit of our function (in our example at *x*→0).

If we consider the set of all rectangles we obtain by shifting this rectangle by adding an arbitrary number to *x*, we get

Such sets one-to-one corresponds to the value of the limit of our function (at *x*→0): Knowing such the set, we can calculate the limit (take its arbitrary element and get its so to say *y*-limit point) and knowing the limit value (*y*), we could write down the definition of this set.

So we have a formula for *generalized limit*:

lim_{x→a} *f*(*x*) = { ν∘*f*|_{Δ(a)}∘*r* ∣ *r*∈*G* }.

where *G* is the group of all horizontal shifts of our space **R**, *f*|_{Δ(a)} is the function *f* of which we are taking limit restricted to the infinitely small interval Δ(*a*) around the point *a*, ν∘ is “stretching” our function graph into the infinitely thin “strip” by applying a topological operation to it.

What all this (especially “infinitely small”) means? Read my writings such as [1].

Why do we consider all shifts of our infinitely small rectangle? To make the limit not dependent on the point *a* to which *x* tends. Otherwise, the limit would depend on the point *a*.

Note that for discontinuous functions elements of our set (our limit is a set) won’t be infinitely small “rectangles” (as on the pictures), but would “touch” more than just one *y* value.

The interesting thing here is that we can apply the above formula to* every* function: for example to a discontinuous function, Dirichlet function, unbounded function, unbounded and discontinuous at every point function, etc. In short, the generalized limit is defined for every function. We have a definition of limit for every function, not only a continuous function!

And it works not only for real numbers. It would work for example for any function between two topological vector spaces (a vector space with a topology).

Hurrah! Now we can define the derivative and integral of *every* function.

Now read [2] for a formal treatment of the above (even for multivalued functions and even wider cases) and also how to put a derivative of a non-differentiable function into a differential equation (that’s interesting, it builds an infinite hierarchy of infinities/singularities and defines topological structures on this hierarchy).

**[1] **Victor Porton. Algebraic General Topology. volume 1. edition 1. At https://mathematics21.org/algebraic-general-topology-and-math-synthesis/, 2019.

**[2] **Victor Porton. Limit of a discontinuous function. At https://mathematics21.org/limit-of-discontinuous-function/, 2019.

In my book Limit of a Discontinuous Function I defined the generalized limit defined for every (even discontinuous) function.

The definition of my generalized limit uses the concept of funcoids. Funcoids are a little advanced topic, what somehow hinder understanding of generalized limit by other mathematicians.

But in the special case of a function *p* -> *q* where *p* is a vector topological space (for example, a normed space, Banach space, Hilbert space, etc.) and *q* is a compact topological space, we can equivalently define the generalized limit without using the concept of funcoids.

First, let me remind the definition of ultralimit as defined in the “standard” nonstandard analysis:

For convenience, the set of filters will include the improper filter.

We can define the image by a function *f* of a filter *X* as the minimal filter that contains all images of elements of *X*.

Ultralimit of a function *f* regarding an ultrafilter (I allow any ultrafilter here, even a principal one, for convenience) *a* is the limit point of the image of *a* by *f*.

If the space *q* is compact, then then the ultralimit always exists because the image of *a* by* f* is also an ultrafilter and the well-known fact in a compact space every ultrafilter has a limit point.

The definition of generalized limit to a compact space without using funcoids follows:

Let *a* be some point of the space *q*. Consider the function *t* that maps every ultrafilter convergent to *a* into its ultralimit. (Exercise: define the corresponding concept for a limit of a sequence.)

It remains the last step: consider the set of compositions of *t* with all possible shifts in our vector space *a*. This is what I call the generalized limit.

Why these looking random operations? Read my book.

This is equivalent to the definition of the generalized limit in my book, because funcoid is determined by its values on ultrafilters.

The generalized limit has the following properties:

- It exists for every function (in our special case with the requirement of compactness of the destination space, but the equivalent definition in my book does not require compactness).
- The limit at a point in the usual sense is determined by the generalized limit (because we include the principal ultrafilter determined by
*a*). - Having a function on the space
*b*we can extend this function to a function acting on values of generalized limit (exercise). - The operations on the values of generalized limits follow the usual algebra laws. For example if the space
*b*is real numbers or complex numbers or a vector space, then*y*–*y*= 0 for every value*y*of the generalized limit.

So far, we defined the generalized limit to compact spaces. See my book Limit of a Discontinuous Function for a generalized case if *b* is not necessarily compact.

Having the generalized limit, we can *easily* define the derivative of an arbitrary function or the definite integral of an arbitrary function!

See my book on how to define nondifferentiable solutions of differential equations. It is a new science.

]]>AGT isn’t new math that’s intended to replace existing mathematical theory, but rather a novel way to express mathematical concepts which were previously inexpressible. I indulge these ideas because I believe, as is the modus operandi for all academics, that any worthwhile, logically consistent idea must be pursued until it is falsified or verified for good.

My AGT and language are admittedly novel, but allow me and will allow mathematicians to speak of mathematical ideas that couldn’t be spoken of before. These ideas are based on the concept of filters that, incidentally, also allows me to develop AGT.

*AGT is a purely intellectual pursuit because it isn’t tainted with the obsession that institutional knowledge must exist as a continuum. That idea is limited, short-sighted and not true to the spirit of mathematical theorization or the pursuit of knowledge.*

In a rudimentary sense, filters are relations we apply to existing sets to express otherwise inexpressible statements. For example, expressing the infinitely small neighborhood around 0 is impossible, but it might be possible if we work with filters. Filters effectively allow us to refer to infinitesimally small or infinitely large sets and conduct mathematical analysis to develop valuable insights.

I am not concerned filters through, which are a fairly well-known idea in mathematics— I am more interested in Filtrators, which is my sole contribution to existing topological theory. Both filters and Filtrators are one of the core components of Algebraic General Topology.

I define a filtrator as:

A pair (B*, *Z) of a poset B and its subset Z ⊆ B

Where

B is the *base *of the filtrator

Z the *core *of the filtrator

Some other definitions of note are:

- I will call a
*lattice filtrator*a pair (B*,*Z) of a lattice B and its subset Z ⊆ B. - I will call a
*complete lattice filtrator*a pair (B*,*Z) of a complete lattice A and its subset Z ⊆ B. - I will call a
*central filtrator*a filtrator (B*, Z*(B)) where*Z*(B) is the center of a bounded lattice A. - I will call an
*element*of a filtrator an element of its base.

To quote Fermat, “*I have discovered a truly remarkable proof of this theorem which this margin is too small to contain*”. I too, like Fermat, find myself constrained for want of space to discuss my entire theory but these are some of the more basic definitions I introduce. Defined this way, it’s possible to express sets like the infinitesimally small set on the real line around 0 and many other similar ideas, which may have great value for mathematical research.

If you’re interested and willing to consider these ideas as a purely intellectual pursuit, you should look up on my Algebraic General Topology book Volume 1. It goes at length into Algebraic General Topology, the basic concepts involved and a collection of all the definitions used to construct the AGT. I am certain, you will find it a joyful exercise.

Leanr more about general topology monograph here.

]]>Algebraic general topology – what is it?

First, what is a general topology? General topology is the theory of topological spaces, as well as uniform spaces, proximity spaces, and metric spaces.

I made a rather big discovery – a general theory that generalizes all these types of spaces in one algebra. That is, instead of four theorems about each of these spaces, we get one.

I called my theory “algebraic general topology”. Algebraic, because instead of the nightmare confusion of logical reasoning from the traditional general topology, we have simple algebraic formulas. Visit my site where you can find these formulas.

I discovered the first formula of algebraic general topology when I was in my first year at the Faculty of Mathematics. For several months I thought, removing unnecessary axioms, and suddenly I understood this first formula. Now I call this formula the definition of a funcoid. That is, funcoids are determined by just one axiom. Again, go to my site.

Then I added what I call reloids – just filters on the Cartesian product of two sets.

It turns out a very interesting theory: funcoids, reloids, connections between them, their generalizations, etc. Much more interesting than the traditional general topology.

Funcoids and reloids can be combined. The resulting operation “composition” (named by analogy with the composition of functions) is associative. Such a structure is called a semigroup. This is an algebraic structure and the properties of funcoids, such as, for example, continuity are described by algebraic formulas. Moreover, this also works, for example, for multi-valued functions.

Not
so long ago, I realized that, moreover, *any
*spaces are actions of
ordered semigroups. Even funcoids and reloids are just a special case
of my new theory. What is funny, before me there was not a single
article on the Internet about actions of ordered semigroups.

Combining
elements such semigroups can write the formula of what I call the
generalized
limit.
This “enhanced”
limit can be applied to *any
*function,
even discontinuous. Don’t
interrupt
viewing of this video: the limit of the discontinuous function does
not exist – that means the author is a pseudo-scientist. I am
talking about a *generalized
*limit,
not the
usual one.

So in my system, any function has a limit, which means that any function has a derivative and an integral. Did you hear what I said? You must be in shock. Moreover, I also gave a definition of non-differentiable solutions of differential equations. This is called the phrase “science of the future”.

All information on algebraic general topology and the limit of a discontinuous function, with all the proofs, is available in the books on my site.

And everything is explained so that even a novice student could understand.

So, read my books and find out, finally, about the analysis on filters. Type in Google algebraic general topology. Do not forget to like this video: support the science of the future. Send money and Bitcoins. But most importantly, read my books.

Learn more about Algebraic General Topology here.

]]>I feel that when the various fields of intellectual endeavor become obsessed with upholding tradition, they stifle room for original thought. The assumption that existing knowledge is complete and only linear, logical processes are worthy of examination, eliminates discourse buries novel, equally as valid ideas into obscurity. This absence of self-reflection is holding back new perspectives in the field of mathematics and restricting budding mathematicians from making valuable contributions.

Sometimes, I wonder how far back we might have been if Hardy hadn’t given Ramanujan a shot at formal mathematics because Ramanujan wasn’t formally trained. The Hardy-Ramanujan collaboration set the grounds for the future of discrete mathematics and proof-theory as we know it today.

It is always the novel idea that reinvigorates mathematical inquiry and it’s the fresh mind, not corrupted by institutional limitations, that pushes math forward. We need an intellectual revolution because conventional topological theory is unlikely to help us resolve the most controversial theorems in mathematics.

Anyone who’s dabbled in mathematical topology knows that there are plenty of unanswered questions and these questions have stood for decades without an answer. One particular problem comes to mind—the Borel Conjecture.

Emile Borel once stated that *every strong measure zero set of reals is countable. *Borel’s statement was established as completely independent of any other axioms set forth in the Zermelo-Fraenkel set theory, which serves as the standard set-theoretical foundation. The particularities of the conjecture are unimportant but the significance of independent streams of mathematical thought is not.

I speak of these instances because my work also lies in the realm of independent mathematical thought that isn’t logically motivated by any other existing mathematical theory. My Algebraic General Topology is a new perspective on the ways that we can speak of mathematical concepts.

Many mathematical theorems and lemmas used in proof theory, like Zorn’s lemma, are introduced when it’s impossible to prove certain theorems explicitly. The application of Zorn’s Lemma to set the proof for the Hahn-Banach theorem represents an event in mathematical history where a mathematical statement, separate from the proof itself was applied to finalize the proof. This, in principle, was the use of a novel approach to resolving a major problem with functional analysis.

My Algebraic General Topology uses a combination of existing literature and some new definitions to explore topological problems that have never been used before. Potentially, when applied to problems like the Hilbert-Smith conjecture, the Borel Conjecture or any other long-standing mathematical problem, my theory might actually represent a solution.

If you’re interested in Algebraic General Topology or wish to discuss your topological ideas, you should read my mathematical research book on AGT called Algebraic General Topology Volume 1. I’m always open to new ideas and thoughts on my work and welcome open debate on any issue you find.

Learn more about general topology monograph here.

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