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Here I present my further research plans in Algebraic General Topology which is a new field of math developed by me. This research plan is not formal and may contain vague statements.
First, there are some open problems (conjectures) in AGT (PDF). These should be solved. Most of these problems are also available in Open Problems Garden for discussion.
Probably Stone duality can be used to transfer some of these open problems into problems about topological spaces, what may help to solve them.
[NEW] On the definition of compact funcoids in my blog.
There must be defined compact funcoids.
Compactoid and compact filters - this article can serve as the idea of how to define compact funcoids.
A compact funcoid can be defined as follows: a funcoid f is compact iff for any filter F if F intersects im f then exists a such that {a}[f]F.
Conjecture: A compact funcoid corresponds to a unique reloid. Probably here can be a stronger conjecture about compact topological spaces.
For compact funcoids the Cantor's theorem that a function continuous on a compact is uniformly continuous.
Algebraic General Topology which is currently a field of point-set topology should be generalized to pointless topology:
We can then define relation between two lattice elements (sic!) as a funcoid between these elements such that both this funcoid and its inverse are complete funcoids. (Well, this is already known under the name adjunction between posets.)
After this we can define reloid on the lattice as a filter on the set of relations between lattice elements.
Or maybe it is equivalent to defining reloids as filters on the set of funcoids - this conjecture needs to be further researched (first formalized then solved).
I have set this wiki to write about pointfree version of AGT there.
Define and research a generalization of funcoids, (n-ary) multidimensional funcoids (by analogy with n-ary relations), both simple case of finitary funcoids and harder case of infinitary funcoids.
Apply the theory of multidimensional funcoids for defining and researching operations (such as arithmetic operations) on the values of limits of discontinuous functions.
It should be defined direct product of funcoids (both finitary and infinitary). It should be a case of product in category theory.
Analog of order topology for funcoids/reloids.
If lim f=f(x) then f is continuous in x (for funcoids and reloids).
If a morphism converges to a value on two sets, it converges to the value also on their union.
There can be straightforwardly defined multidimensional reloids (as a filter on the set of relations of several variables). A n-ary funcoid can be defined as n-ary relation on the set of subsets of a set such that it is distributive regarding set union on each variable and is false if any variable is empty set.
This set of issues is not complete, there are yet much other things to research about funcoids and reloids.
The generalization of the theorem that uniform continuity implies proximity continuity which in turn implies topological continuity.
I expect that the following will be a common generalization of reloids and funcoids: a complete lattice filtrator whose base is the set of binary relations (on some set). However now is not enough known about the filtrator of funcoids, e.g. whether it is finitely meet-closed.
A set is connected if every function from it to a discrete space is constant. Can this be generalized for generalized connectedness and generalized continuity? I have no idea how to relate these two concepts in general.
Develop theory of funcoidal groups by analogy with topological groups. Attempt to use this theory to solve this open problem.
Research what I call micronization, a mapping from a "global" funcoid (such as a partial order) to a "local" funcoids (such as a topology).
Copyright © 2004-2011 Victor Porton. All rights reserved.
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