Funcoid
From Wikinfo
Funcoids are topological objects and are a generalization of proximity spaces.
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Definition
In this article filter objects refers to filter objects on a set.
A funcoid is a pair of functions (a;b) where both a and b are functions mapping the set of filter objects to itself such that Y∩a(X)=∅⇔X∩b(Y)=∅ for any filter objects X and Y.
By definition 〈(a;b)〉 = a.
The relation [f] for a funcoid f is defined by the formula A[f]B ⇔ 〈f〉A ∩ B ≠ ∅ where A and B are filter objects.
Basic properties
See Funcoids and reloids for details.
Inverse funcoid of funcoid (a;b) is (a;b)-1 = (b;a).
Composition of funcoids is defined by the formula (a2; b2)∘(a1; b1) = (a2∘a1; b1∘b2).
The following properties hold for any funcoids f, g, h:
- (f-1)-1 = f;
- f∘(g∘h) = (f∘g)∘h;
- (f∘g)-1 = g-1∘f-1.
Funcoids and proximities
Every funcoid bijectively corresponds to a certain binary relation δ on sets which generalized a proximity.
See Funcoids and reloids.
More to add.
Relation with reloids
Funcoids are closely related with reloids. For example to any reloid f corresponds the funcoid (FCD)f.
See Funcoids and reloids for details.
About discovery of funcoids
During decades general topology was considered a "dead" field of research (that is it was assumed that no significantly new can be discovered here).
Discovery of funcoids by Victor Porton gives a fresh stream of new research in general topology.
At the time of writing this there are more than 17 open problems about funcoids and reloids, see Open Problem Garden.
Victor Porton pleads to nominate him for Abel Prize for the discovery of funcoids.
References
External references
- Algebraic General Topology - the theory of funcoids and reloids.
