About myself
I’m not a professional mathematician, I work as a pro-grammer.
I have been studying in a university in Russia but have notfinished my study.
So, I know little beyond my specialization.
Nevertheless in my free time I discovered a new theory whichwould completely overturn general topology.
About this lesson
In this lesson I present my discovery, the theory of funcoidsand reloids.
I will not give here proofs of my results, as you can read myactual articles if you get interested in knowing the details.
The motivation for study of funcoids and reloids is thatthey are an elegant generalization of “spaces” (topological,pretopological, proximity, uniform spaces) and of binaryrelations between elements of spaces.
For brevity I will be sometimes a little informal in this lesson,for instance considering composition of funcoids (see below)without explicitly formulating that they are composable.
What is Algebraic General Topology?
I have introduced and researched objects called funcoids,reloids, and their generalizations.
I have named the theory of these objects Algebraic GeneralTopology.
See My generalizations
Usage of funcoids and reloids
That funcoids and reloids are a common generalization ofspaces and functions (functions are a special case of binaryrelations), it makes them a smart tool for expressing prop-erties of functions in regard of spaces.
For example, the statement “f is a continuous function froma space µ to a space ν” can be expressed by the formula: Algebraic General Topology is a generalization of customarygeneral topology but is much more elegant than the cus-tomary general topology.
The theory of funcoids and reloids is based on the theory offilters.
I’ve written an article on the theory of filters and their gen-eralizations: In that article I consider filters on arbitrary posets and gen-eralizations thereof. But in this lecture we will consider onlyfilters on the lattice of subsets of some set.
Lattices and Filters
In order not to confuse poset/lattice operators with set-the-oretic operators, I will denote partial order as ⊑ and latticeoperators as ⊔, ⊓, For my notation to be consistent, I need to order filtersreverse to set theoretic inclusion of filters. I will denote Fthe lattice of filters (on some set) including the improperfilter ordered reverse to set-theoretic inclusion of filters: More about filters
I will denote Base(A) the set on which the filter A is defined.
I will denote the principal filter on a set A corresponding toa set X as Ultrafilters are atoms of the lattice of filter objects (atomicfilters).
Lattice of filters
The lattice F(A) of reverse ordered filters (on some set A) is: having minimum and maximum 0F(A) and 1F(A) Read more about such lattices and more general posets inmy article: “Filters on Posets and Generalizations” Generalized proximities
The most natural way to introduce funcoids is generalizingproximity spaces.
Let δ be a proximity on a set ℧. It can be extended fromsubsets of ℧ to filters on ℧ by the formula X δ′ Y ⇔ ∀X ∈ X , Y ∈ Y: X δ Y .
I’ve proved that there exist two functions α: F(℧) → F(℧)and β: F(℧) → F(℧) such that Definition of funcoids
Let’s fix two sets A and B.
The pair of two functions α: F(A) → F(B) and β: F(B) →F(A) such that denotes a funcoid . Strictly speaking, a funcoid is a quadruple(A; B; α; β) conforming to the above formula.
Thus funcoids are a generalization of proximity spaces.
I call funcoids (A; B; α; β) funcoids from A to B and denotethe set of funcoids from A to B as FCD(A; B).
Source and destination of a funcoid
The source and the destination of a funcoid f = (A; B; F ) are Components of a funcoid: part 1
Let f = (A; B; α; β ) be a funcoid. Then bydefinition: A funcoid can be inverted (the inverse is also a funcoid): Components of a funcoid: part 2
An important property of funcoids: a funcoid f is completelycharacterized by just one of its components, say f . More-over f is determined by values of f on principal filters.
Funcoids and relations between filters
By definition
and X [f ]∗ Y ⇔ ↑AX [f ] ↑BY .
A funcoid f is completely characterized by the relation [f ]and even by f ∗ or [f ]∗.
Principal funcoids
Let A and B be sets.
For every binary relation F ∈ P(A × B) there exists afuncoid ↑FCD(A;B)F ∈ FCD (A; B) defined by the formula (forevery X ∈ PA) This funcoid is unique because a funcoid is determined bythe values of its first component on principal filters.
I call funcoids corresponding to a binary relation by the for-mula above as principal funcoids.
Funcoids & pretopologies
Let α be a pretopology, so α is a function ℧ → F(℧). Thenthere exists a funcoid f such that (the join is taken on the lattice of filters).
So funcoids are a generalization of pretopologies.
Funcoids & preclosures
Let F be a preclosure (for example, F may be a topologicalspace in Kuratowski sense). Then there exists a funcoid fsuch that Thus funcoids are a generalization of preclosures.
Composition of funcoids
The composition of binary relations induces for principalfuncoids composition which complies with the formulas: We can define composition for funcoids by the same for-mulas. Strictly speaking the composition of funcoids isdefined by the formula: (B; C; α2; β2) ◦ (A; B; α1; β1) = (A; C; α2 ◦ α1; β1 ◦ β2).
h ◦ (g ◦ f ) = (h ◦ g) ◦ f .
Alternate representations of funcoids
Above I defined funcoids as quadruples. But a funcoid canbe represented in two other ways: as a binary relation δ ∈ P(PA × PB) between sets as a function α: PA → F(B) from sets to filters Below I will show the exact conditions required for δ andα in order to represent a funcoid. Funcoids from A to Bbijectively correspond to such δ and α.
Funcoids as binary relations
A binary relation δ ∈ P(PA × PB) corresponds to a fun-coid if and only if it complies to the formulas (for all suitablesets I, J, K): ¬(∅ δ I); I ∪ J δ K ⇔ I δ K ∨ J δ K;¬(I δ ∅); K δ I ∪ J ⇔ K δ I ∨ K δ J.
The funcoid f and relation δ are related by the formulas: X [f ] Y ⇔ ∀X ∈ X , Y ∈ Y: X δ Y ; Funcoids as functions
A function α ∈ PA → F(B) corresponds to a funcoid if and
only if it complies to the formulas (for all sets I , J ∈ PA):
The funcoid f and function α are related by the formulas: Order of funcoids
The set FCD(A; B) of funcoids from A to B is a poset withorder defined by the formula: Moreover it is a complete, distributive, co-Brouwerian, atom-istic lattice.
Values of a join or meet of funcoids
For every R ∈ PFCD(A; B) and X ∈ PA, Y ∈ PB For every R ∈ PFCD(A; B) and x, y being ultrafilters on Aand B correspondingly we have: Funcoidal product
The funcoidal product of filters is a generalization of theCartesian product of sets.
Let A and B be filters. Then there exists a unique funcoid(the funcoidal product of A and B) A ×FCD B such that Restricted identity funcoid
Restricted identity funcoids are a generalization of identityrelations on a set.
Let A be a filter. Then there exists a unique funcoid (therestricted identity funcoid on A) idFCD Restriction of funcoids
Restriction of a funcoid f to a filter A is defined by theformula T0-, T1- and T2-separable funcoids
A funcoid f is T1-separable when
An endofuncoid (a funcoid with the same source and desti-nation) is: 1. T0-separable when f ⊓ f −1 is T1-separable.
2. T2-separable when f −1 ◦ f is T1-separable.
Some properties of funcoids
Let f , g be funcoids, X , Z be filters. Then X [g ◦ f ] Z iffthere exists an ultrafilter y such that X [f ] y and y [g] Z.
Let f , g, h be funcoids. Then f ◦ (g ⊔ h) = f ◦ g ⊔ f ◦ h.
Reloids are a trivial generalization of uniform spaces.
Roughly speaking, a reloid is a filter on a Cartesian productof two sets.
To be precise, I define a reloid as a triple f = (A; B; F ) whereA and B are sets and F is a filter on A × B.
Note that reloids are also a generalization of binary relations.
The reverse reloid f −1 is defined as follows: f −1 = (A; B; F )−1 = (B; A; F −1).
I will also denote GR(A; B; F ) = F .
Principal reloids
Let F be a binary relation between sets A and B.
Then is so called the principal reloid corresponding to the relationF .
Composition of reloids
Let f = (A; B; F ) and g = (B; C; G) be reloids. Thecomposition g ◦ f is defined by the formula ↑RLD(A;C)(Y ◦ X) | X ∈ GR F , Y ∈ GR G .
In other words, the composition corresponds to the filter (onA × C) defined by the base {Y ◦ X | X ∈ GR F , Y ∈ GR G}.
Composition of reloids is associative.
Reloidal product
The reloidal product of filters is a generalization of the Carte-sian product of sets.
Let A and B be filters. Then the reloidal product of A andB is defined by the formula: ↑RLD(A;B)(A × B) | A ∈ A, B ∈ B .
In other words, the reloidal product is the reloid defined bythe base Restricted identity reloid
The identity reloid on a set A is defined as idRLD(A) =↑RLD(A;A)idA.
Similarly to the above defined restricted identity funcoid, wecan also define the restricted identity reloid = idRLD(Base(A)) ⊓ A ×RLD 1F(Base(A)) .
↑RLD(Base(A);Base(A))idA | A ∈ up A .
Restriction of a reloid
Restriction of a reloid f to a filter A is defined by the formula Some properties of reloids
f ◦ (g ⊔ h) = f ◦ g ⊔ f ◦ h.
Ordered and dagger categories
I call a partially ordered category a category together with apartial order on each of its Hom-sets.
A dagger category is a category together with a functionf f † on the set of morphisms which inverses the source and the destination of the morphism and is subject to thefollowing conditions: Categories of funcoids and reloids
Funcoids with objects being sets and composition of funcoids
form a category which I call the category of funcoids.
The same holds for reloids.
Categories of funcoids and reloids are both partially ordered
dagger categories with “the dagger” defined as
Special morphisms
Let f be a morphism of a partially ordered dagger category.
f is monovalued when f ◦ f † ⊑ 1Dst f.
f is entirely defined when f † ◦ f ⊒ 1Src f.
f is injective when f † ◦ f ⊑ 1Src f.
f is surjective when f ◦ f † ⊒ 1Dst f.
It’s easy to show that this is a generalization of monovalued,
entirely defined, injective, and surjective binary relations as
morphisms of the category Rel.
Monovalued funcoids
I will denote “atoms a” the set of atoms under a for an ele-ment a of a poset.
The following statements are equivalent for a funcoid f : 2. ∀a ∈ atoms 1F(Src f): f a ∈ atoms 1F(Dst f) ∪ 0F(Dst f) .
3. ∀I , J ∈ F(Dst f ): f −1 (I ⊓ J ) = f −1 I ⊓ f −1 J .
4. ∀I , J ∈ P(Dst f ): f −1 ∗(I ∩ J) = f −1 ∗I ⊓ f −1 ∗J.
Consequently a principal funcoid is monovalued iff its corre-sponding binary relation is monovalued (a function).
Functions between spaces
Let µ and ν be funcoids corresponding to a (pre)topologicalspaces, or proximity spaces, or µ and ν be uniform spaces(that is reloids).
Let f be the funcoid or reloid corresponding to a functionfrom the first space to the second space.
Continuous morphisms
Continuity, proximal continuity, and uniform continuity off is expressed by the same formula: In the case if f is monovalued and entirely defined, we have f ◦ µ ⊑ ν ◦ f ⇔ µ ⊑ f −1 ◦ ν ◦ f ⇔ f ◦ µ ◦ f −1 ⊑ ν.
This can be generalized for any partially ordered dagger cat-egories.
Relationships of funcoids and reloids
For every sets A, B we have FCD(A; B) and RLD(A; B)interrelated by the below defined functions: The funcoid induced by a reloid
Every reloid f ∈ RLD (A; B) induces a funcoid (FCD)f ∈FCD(A; B) by the following formulas: X [(FCD)f ] Y ⇔ ∀F ∈ GR f : X ↑FCD(A;B)F Y; We have for every composable reloids f and g: (FCD)(g ◦ f ) = ((FCD)g) ◦ ((FCD)f ).
I will skip some minor facts on this topic.
The reloids induced by a funcoid
Every funcoid f ∈ FCD(A; B) induces a reloid from A to Bin two ways, namely intersection of outward relations andunion of inward direct products of filters: {A ×RLD B | A ∈ F(A), B ∈ F(B), A ×FCD B ⊑ f }.
{a ×RLD b | a is an atom of F(A), b is an atom Some Galois connections
(FCD): RLD(A; B) → FCD(A; B) is the lower adjoint of(RLD)in: FCD(A; B) → RLD(A; B). Thus for every set S of reloids or T of funcoids.
Convergence of funcoids
A filter F converges to a filter A regarding to a funcoid µ (F→A) iff F ⊑ µ A. (This generalizes the standarddefinition of filter convergent to a point or to a set.)A funcoid f converges to a filter A regarding to a funcoid µ (f→A) iff im f ⊑ µ A that is iff im f→A.
A funcoid f converges to a filter A on a filter B regarding to We can define also convergence for a reloid f : f→A⇔ im f ⊑ µ A or what is the same f→A ⇔ (FCD)f→A.
Limit of a funcoid
for a T2-separable funcoid µ and a non-empty funcoid f .
It is defined correctly, that is f has no more than one limit.
Generalized limit
We can define a (generalized) limit for an arbitrary (discon-tinuous) function, for example any function on the set ofreals, or more generally from any topological vector space toany topological vector space, etc.
An idea is that the limit should not change when translatingto an other point of the space. Thus we need to fix a groupG of translations (or any other transformations) of our space.
Conditions for a generalized limit
Let µ and ν be funcoids on a set ℧, and G be a group offunctions.
Let D be a set such that ∀r ∈ G: im r ⊆ D ∧ ∀x, y ∈ D∃r ∈ G: r(x) = y.
We require that µ and every r ∈ G commute, that is µ ◦ r =r ◦ µ (for r considered as a principal funcoid).
We require for every y ∈ ℧ Definition of generalized limit
for any funcoid f . (Here G is considered as a set of principalfuncoids.)The limit at a point x is defined as Limits and generalized limits
τ (y)= { µ {x} ×FCD ν {y } | x ∈ D }.
Theorem Let µ be a T2-separable funcoid and ν be a non-
empty funcoid such that ν ⊑ ν ◦ ν. If limx f = y then
xlimx f = τ (y).
This theorem establishes a bijective correspondence (namely
τ ) between limits and a subset of generalized limits.
The skipped topics
In my speech I have skipped the following topics: specifying a funcoid by its values on ultrafilters complete funcoids and reloids, completion of funcoidsand reloids connectedness of sets and of filters regarding funcoidsand reloids Further research directions
I also have researched pointfree funcoids, a generalization offuncoids relevant for pointfree topology, and multifuncoids,its further generalization.
The future research topics include: I have also formulated a quite large number of open problemsrelated to filters, funcoids, and reloids. If you need a researchfield, I suggest you to solve my open problems.
I also have written a book.


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