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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.
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 full funcoids.
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).
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). How to define multidimensional funcoids?
This set of issues is not complete, there are yet much other things to research about funcoids and reloids.
Uniformly connected spaces.
Copyright © 2004-2007 Victor Porton. All rights reserved.
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