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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 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.
Algebraic General Topology. Volume 1 (PDF, draft) See also development version (undergoes heave rewrite and may be inconsistent) and old version formatted with TeXmacs. My theory as a book. You are strongly recommended to read this book, not "legacy articles" below. (I am going to rewrite this book.)
Sets of filters, of ideals, and yet two kinds of objects are isomorphic.
Funcoids bijectively correspond to filters on a certain lattice.
I define a categorical direct product in the category of continuous maps between endofuncoids and some other similar categories.
I define a equalizers and co-equalizers in certain categories.
Compactness of funcoids generalizing compactness of topological spaces is defined.
I define an order embedding from the set of frames to the set of pointfree endo-funcoids.
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.
I consider some categories related with pointfree funcoids.
I consider Cauchy filters on reloids, generalizing Cauchy filters on uniform spaces. Using Cauchy filters, I define Cauchy-complete reloids, generalizing complete uniform spaces.
Staroids on powersets described as filters on certain lattices.
A new idea about staroids. A very preliminary draft.
A proof that funcoids form a co-frame (without axiom of choices).
Introduced another version of cross-composition of funcoids. This forms a category with star-morphisms. It is conjectured that this category is quasi-invertible.
If you solve any of these problems, please notify me! Also (half jokingly) read here
Reduced limits (PDF, very rough partial draft) - Operations with values of singularities. Conjectures about application of this to General Relativity and black holes.
Wiki about proving certain categories are cartesian closed
Certain categories are cartesian closed (rough draft with errors)
Open Problems in AGT (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 (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
Research Plans and Ideas in Algebraic General Topology
Correct errors in Compact funcoids.
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 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.
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.
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.
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.
Consider a generalization of funcoids.
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.
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.
A generalization of totally bounded uniform spaces. It splits into several distinct concepts which are equivalent for the special case of uniform spaces.
It's about filters on arbitrary lattices and posets as well as certain generalizations thereof.
Some easy strengthening of some theorems from the article
Filters on Posets and Generalizations
.
Partially solved problem 1
from Filters on Posets and Generalizations.
In this short article I investigate properties of free stars, first introduced in this article.
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.
Todd Trimble's commentary/notes on my work (an alternative view on my theory) (see also my response on Todd's notes)
Quasi-cartesian functions - a blind valley of research, a theory which I am not going to publish.
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:
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.
Algebraic General Topology at Facebook.
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Algebraic General Topology at WikInfo.
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