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

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

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