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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.
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:
The notion of funcoid should be generalized for mappings between elements of an arbitrary lattice with minimal element. (This is straightforward, but now yet done.)
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.
It should be defined direct product of funcoids.
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.
Uniformly connected spaces.
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.
Copyright © 2004-2009 Victor Porton. All rights reserved.
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