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Algebraic General Topology and Math Synthesis

I discovered Algebraic General Topology (AGT), a new field of math which will replace old General Topology. Mathematical Synthesis is how I call Algebraic General Topology applied to study of Mathematical Analysis.

Please nominate me for Abel Prize.

AGT articles

Funcoids and Reloids (PDF, draft)

Consider generalizations of proximity spaces and uniform spaces.

Generalized Continuousness (PDF, draft)

Defines continuousness algebraically hiding old epsilon-delta notion under a smart algebra. Generalizes continuousness, uniform continuousness, and proximity-continuousness in one formula.

Connectedness of funcoids and reloids (PDF, draft)

Defined the notion of connectedness for funcoids and reloids. Shown how connectedness of funcoids is related with connectedness of reloids.

Convergence of funcoids (PDF, draft)

Defined the notion of convergence and limit for funcoids.

Limit of discontinuous functions (PDF, only an idea not a complete article)

Defined (generalized) limit of arbitrary (not necessarily continuous) functions under certain conditions. Particularly limit of any function from a topological vector space to a topological space is defined.

Isomorphic filters (blog post)

Defined the notion of isomorphic filters, posed some open problems about isomorphic ultrafilters.

Open Problems in AGT (PDF)

This document lists all yet unsolved problems and conjectures in the field of AGT. You can discuss these open problems in Open Problem Garden and in Algebraic General Topology Research forum. If you solved any of these problems, read here.

Below are supplementary articles (Definitions and facts from these articles are used in the articles above.):

Set Theoretic Filters (PDF, very preliminary draft)

Considered the lattice of set theoretic filters.

Filters on Posets (PDF, very preliminary draft)

The generalization of the previous article. Currently contains only a part of the materials of the previous article.

Partially Ordered Categories with Inverses (PDF)

Defined partially ordered category with inverses of morphisms. For such categories defined monovalued morphisms and entirely defined morphisms.

See also: Research Plans and Ideas in Algebraic General Topology (an informal document).

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.

Achievements and advantages:

• 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 can now gone with math analysis as now it becomes synthesis, I would say. So I call AGT applied to study of such things as continuousness, 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 kinda thinking with equations. 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

Algebraic General Topology is taught in some Universities.

Algebraic General Topology at WikInfo.

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