**Algebraic General Topology. Vol 1**:
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**Axiomatic Theory of Formulas**:
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About myself

I’m not a profe s sional mathematician, I work as a pro-

grammer.

I have been studying in a university in Russia but have not

ﬁnished my study.

So, I know little beyond my specia lization.

Neverth eless in my free time I discovered a n ew theory which

would com pletely overturn gen er al topology.

1

About this les son

In this lesson I present my discovery, the theory of funcoids

and reloid s .

I will not give here proofs of my results, a s you can read my

actual ar t icles if you get interested in know in g the details.

The motivation for study of funcoids and reloi ds is th at

they are an elegant generalization of “spaces” (topological,

pretopological, proximity, uniform spaces) and of binary

relations between ele ments of spaces.

For brevity I will be sometimes a li ttle inf o rmal in this lesson,

for instance co nsidering compositi o n of fun coids (see below)

withou t explicitly formulating that they a re compo s able.

2

What is Algebraic General Topology?

I have introduced and r es earched objects called fun coids,

reloids, and their generalizations.

I have named the theor y of these objects Algebraic Genera l

Topology.

See

http://www.mathematics21.org

3

My gener alizati ons

Funcoids are a gener a lization of:

• proximity spaces

• pretopological spaces

• preclos ures

• digraphs (that is bin a ry relations)

Reloids are a generalization o f:

• unifor m spaces

• digraphs (that is bin a ry relations)

4

Usage of f uncoids and reloids

That funcoid s and reloids are a common ge neralization of

spaces and functions (functions are a special case of b in ary

relations), it makes the m a smart tool for expressing prop-

erties of fun ct ions in reg a rd of spaces .

For example, the stateme nt “f is a continu o us function from

a space µ to a space ν” can be expressed by the formula:

f ◦ µ ⊑ ν ◦ f .

Algebraic General Topolog y is a generalization of customary

general topology but is mu ch more elegant than the cus-

tomary general topology.

5

Filters

The theory of funcoids and reloids is based on the th eory of

ﬁlters.

I’ve written an article on the theory of ﬁlters and their gen-

eralizations:

http://www.mathematics21.org

In that article I consid er ﬁlters on arbit rary posets and gen-

eralizations thereof. But in this lecture we will cons id er only

ﬁlters on the latt ice of subs et s of some set.

6

Latti ces and Filter s

In order not to con fuse poset/lattice ope rators with set-the-

oretic o per a tors, I will denote partial order a s ⊑ and lattice

operat ors as ⊔, ⊓,

F

,

d

.

For my notation to be consist ent, I need to order ﬁlters

reverse t o set th eoretic inclusion of ﬁlters. I will deno te F

the lattice of ﬁ lt ers (o n some set ) i ncluding the impr o per

ﬁlter orde red reverse to set- theoretic inclusion of ﬁlters:

A ⊑ B ⇔ A ⊇ B.

7

More about ﬁlters

I will denote Base(A) the set on whi ch the ﬁlter A is deﬁned.

I will denote the p ri ncipal ﬁlter on a set A corresponding to

a set X as

↑

A

X.

Ultraﬁlters are atoms of the lattice of ﬁlte r objects ( a tomic

ﬁlters).

8

Latti ce of ﬁlters

The lattice F(A ) of reverse ordered ﬁlter s (on som e set A) is:

• having mi nimum and maximu m 0

F(A)

and 1

F(A)

• atomi s tic

• complete

• distribu tive

• co-Bro uwerian (A ⊔

d

S =

d

{A ⊔ X | X ∈ S })

Read mo re abou t su ch lattices and more general po s et s i n

my article:

“Filters on Posets and Generaliz ations”

9

Generalized proximities

The most natu ral way t o intr od uce func o ids is gener a lizing

proximity spaces.

Let δ be a proximity on a se t ℧. It can be extended from

subsets of ℧ to ﬁlt ers on ℧ by the for mula

X δ

′

Y ⇔ ∀X ∈ X , Y ∈ Y: X δ Y .

I’ve pr oved that t here exist two functi o ns α : F(℧) → F(℧)

and β: F(℧) → F(℧) such that

X δ

′

Y ⇔ Y ⊓α X

0

F(℧)

⇔ X ⊓ β Y

0

F(℧)

.

10

Deﬁnit ion 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

Y ⊓ α X

0

F(B)

⇔ X ⊓ β Y

0

F(A)

denotes a funcoid. Strictl y speaking, a funcoid is a quad ruple

(A; B; α ; β) conforming to the above formula.

Thus func oids are a gener alization of proxim ity spac es.

I call funcoids (A; B; α; β) funcoids fro m A to B and denote

the s et of funco id s from A to B as FCD(A; B).

11

Source and destination of a f uncoid

The source an d the destination of a funcoi d f =(A; B; F ) are

Src f = A; Dst f = B.

12

Components of a funcoid: part 1

Let f = (A; B; α; β) be a fu ncoid. The n by

deﬁnition:

hf i = α.

A funcoid can be inverted (the invers e is al s o a fun coid):

f

−1

= (B; A; β; α).

13

Components of a funcoid: part 2

We h ave

hf

−1

i = β.

Thus a funcoid f ha s two component s :

hf i = α and hf

−1

i = β.

An important property of funcoids : a funcoid f is completely

characterized by jus t one of its compone nts , s ay hf i. M ore-

over f is determined by values of hf i on p rincipal ﬁlters.

14

Funcoids and r elations between ﬁlters

By deﬁnition

X [f ] Y ⇔ Y ⊓ hf iX

0

F(B)

.

We h ave

X [f ] Y ⇔ Y ⊓ hf iX

0

F(B)

⇔ X ⊓ hf iY

0

F(A)

.

For brevity I will als o deﬁne:

hf i

∗

X = hf i↑

A

X a nd X [f ]

∗

Y ⇔ ↑

A

X [f ] ↑

B

Y .

A funcoid f is completely characterized by the relation [f]

and even by hf i

∗

or [f ]

∗

.

15

Principal funcoids

Let A and B be sets.

For every bin a ry relat ion F ∈ P(A × B) the re exi s ts a

funcoid ↑

FC D(A;B)

F ∈ FCD (A; B) d eﬁ ned by th e formula (for

every X ∈ PA)

h↑

FCD(A;B)

F i

∗

X = ↑

B

F [X].

This funco id is unique because a funco id is determined by

the values of its ﬁrst component on princ ip al ﬁlters.

I call funcoids corresponding to a binary relat io n by the for-

mula above as principal funcoids.

16

Funcoids & pre topol ogie s

Let α be a p r etopology, so α is a function ℧ → F(℧). Th en

there exists a funcoid f such that

hf i

∗

X =

G

{α(x) | x ∈ X }

(the join is taken on the lattice of ﬁlters).

So funcoids are a gene ralization of pretopologies.

17

Funcoids & pre closures

Let F be a precl o s ure (for example, F may be a topological

space in Kuratowski sense). Then there exists a funcoid f

such that

hf i

∗

X = ↑

B

F (X).

Thus func oids are a gener alization of preclosures.

18

Composi tion of funcoids

The compositio n of binary rel ations induces for principal

funcoids compo s ition wh ich complies wi th the formu la s :

hg ◦ f i = hgi ◦ hf i a nd h(g ◦ f)

−1

i = hf

−1

i ◦ hg

−1

i.

We can deﬁ ne composition for funcoids by the same for-

mulas. Strictly speaking the compositio n of funcoids is

deﬁned by the formula:

(B; C; α

2

; β

2

) ◦ (A; B; α

1

; β

1

) = (A; C; α

2

◦ α

1

; β

1

◦ β

2

).

Composition of funcoids is assoc iative:

h ◦ (g ◦ f) = (h ◦ g) ◦ f.

19

Al ternate represe ntations of funcoids

Above I deﬁned fun co ids as quadruples. But a funco id can

be represented in two other ways:

• as a binary relat ion δ ∈ P(PA × PB) between s et s

• as a function α: PA → F(B) from set s to ﬁlters

Below I will show the exact conditions required for δ an d

α in order t o represent a funcoid. Funco ids from A t o B

bijective ly correspond to such δ and α.

20

Funcoids as binary relations

A binary relation δ ∈ P(PA × PB) corresponds to a fun-

coid if and only if it com plies to the formu las (for all su itable

sets 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 r elation δ are related by the fo rmulas:

X [f ] Y ⇔ ∀X ∈ X , Y ∈ Y: X δ Y ;

X δ Y ⇔ X [f ]

∗

Y .

21

Funcoids as functions

A function α ∈ PA → F(B) correspon ds to a f un coid if a nd

only if it complies to the formulas (for all sets I , J ∈ PA):

α ∅ = 0

F(B)

; α(I ∪ J) = αI ⊔ αJ.

The funcoid f and f unction α ar e re la ted by the formulas:

hf iX =

l

{αX | X ∈ X };

αX = hf i

∗

X.

22

Order of funcoids

The set FCD(A; B) of funcoids from A to B is a poset with

order de ﬁned by t he formula:

f ⊑ g ⇔ [f ]⊆[g].

More over it is a complete, di stributive, co-Brouwerian, ato m -

istic lattice.

23

Val ues of a join or meet of funcoids

For every R ∈ PFCD(A; B) and X ∈ PA, Y ∈ PB

1. X [

F

R]

∗

Y ⇔ ∃f ∈ R: X [f]

∗

Y ;

2. h

F

Ri

∗

X =

F

{hf i

∗

X | f ∈ R}.

For every R ∈ PFCD(A; B) and x, y being ultraﬁlters on A

and B correspond in g ly we have:

1. x [

d

R] y ⇔ ∀ f ∈ R: x [f ] y;

2. h

d

Rix =

T

{hf ix | f ∈ R}.

24

Funcoidal product

The funcoidal product of ﬁlters is a generalization of the

Cartesian product of sets.

Let A and B be ﬁlters. Then there ex ists a uniq ue funcoid

(the funco i dal product of A and B) A ×

FCD

B such that

hA ×

FCD

BiX =

(

B if X ⊓ A

0

Base(A)

;

0

Base(B)

if X ⊓ A = 0

Base(A)

;

X [A ×

FCD

B] Y ⇔ X ⊓ A

0

Base(A )

∧ Y ⊓ B

0

Base(B)

.

25

Restricted identity funcoid

Restricted identity fun coids are a generalization o f identity

relations on a set.

Let A be a ﬁlter . T hen there exists a unique f uncoid (the

restri cted identity funcoid on A) id

A

FC D

such that:

hid

A

FCD

iX = X ⊓ A;

X [id

A

FCD

] Y ⇔ X ⊓ Y ⊓ A

0

Base(A)

.

26

Restriction of funcoids

Restriction of a funcoid f to a ﬁlte r A is deﬁned by the

formula

f |

A

=f ◦ id

A

FCD

.

It follows

f |

A

=f ⊓ (A ×

FCD

1

F(Dst f )

).

27

T

0

-, T

1

- and T

2

-separable funcoids

A funcoid f is T

1

-separable when

∀α ∈ Src f , β ∈ Dst f : (α

β ⇒ ¬ ({α} [f ]

∗

{β })).

An endofun coid (a funco id with the same source and desti-

nation) i s :

1. T

0

-sepa rable when f ⊓ f

−1

is T

1

-sepa rable.

2. T

2

-sepa rable when f

−1

◦ f is T

1

-sepa rable.

28

Some properties of funcoids

Let f, g be f uncoids, X , Z be ﬁ lt ers. Then X [g ◦ f] Z iﬀ

there exists an ultraﬁlter y su ch th at X [f] y and y [g] Z.

Let f, g, h be funcoids. Then

f ◦ (g ⊔ h) = f ◦ g ⊔ f ◦ h.

Also

f ⊓ (A ×

FCD

B) = id

B

FCD

◦ f ◦ id

A

FCD

.

29

Reloids

Reloids are a trivi a l generalization of uniform spaces.

Roughly spea king, a relo id is a ﬁlter on a Cart es ian produ ct

of two se ts.

To be precise, I deﬁne a reloid as a triple f = (A; B; F ) where

A and B are sets and F is a ﬁlter on A × B.

Note that reloids are also a generaliza ti on of binary rel a tions.

The reverse reloid f

−1

is deﬁned as follows:

f

−1

= (A; B; F )

−1

= (B; A; F

−1

).

I will also deno te GR(A; B; F ) = F .

30

Principal reloi ds

Let F be a bin a ry relat io n be tween sets A and B.

Then

↑

RLD(A;B)

F = (A; B; ↑

A×B

F )

is so called the principal reloid c orrespo nding to t he relation

F .

31

Composi tion of rel oids

Let f = (A; B; F ) a nd g = (B; C; G) be reloids. The

composition g ◦ f is deﬁned by the for mula

g ◦ f =

l

↑

RLD(A;C)

(Y ◦ X) | X ∈ GR F , Y ∈ GR G

.

In other words, t he composition corresponds to t he ﬁlter (on

A × C) deﬁn ed by the b ase

{Y ◦ X | X ∈ GR F , Y ∈ GR G}.

Composition of reloids is associative.

32

Reloidal product

The reloidal product of ﬁlter s is a generalization of the Carte-

sian prod uct of s ets.

Let A and B be ﬁlters. Th en the reloid al prod uct of A a nd

B is deﬁned by the formu la:

A ×

RLD

B =

l

↑

RLD(A;B)

(A × B) | A ∈ A, B ∈ B

.

In other words, the reloidal product is th e r eloid deﬁned by

the base

{A × B | A ∈ A, B ∈ B}.

33

Restricted identity reloid

The identity reloid on a set A is deﬁned as id

RLD(A)

=

↑

RLD(A;A)

id

A

.

Similarly to the above deﬁned restricted ide ntity funcoid, we

can also deﬁne the restricted ident i ty reloid

i d

A

RLD

= id

RLD(Base(A ))

⊓

A ×

RLD

1

F(Base(A))

.

We h ave

id

A

RLD

=

l

↑

RLD(Base(A);Base(A))

id

A

| A ∈ up A

.

34

Restriction of a reloid

Restriction of a reloid f to a ﬁlter A is deﬁned by the fo rmula

f |

A

=f ◦ id

A

or

f |

A

=f ⊓ (A ×

RLD

1

F(Base(A))

).

35

Some properties of reloids

Let f, g, h be reloids. Then

f ◦ (g ⊔ h) = f ◦ g ⊔ f ◦ h.

Also

f ⊓ (A ×

RLD

B) = id

B

RLD

◦ f ◦ id

A

RLD

.

36

Ordered and dagger categories

I call a partially ordered category a category together with a

partial order on each of its Hom-sets.

A d agger category is a category together with a function

f

f

†

on the set of morphisms which inver s es the source

and th e destination of the morphism and is subject to the

following cond it io ns:

1. f

††

= f ;

2. (g ◦ f)

†

= f

†

◦ g

†

;

3. (1

A

)

†

= 1

A

.

37

Categor ies of funcoids and reloids

Funcoids with objects being sets and compositio n of f uncoids

form a category which I c a ll t he categ ory of funcoids.

The same holds for reloids.

Categories of funcoids and reloids are bo th pa rtially ordered

dagger catego ries with “th e dagger” deﬁned as

f

f

−1

.

The orde r and the dag ger agree:

f

†

⊑ g

†

⇔ f ⊑ g.

38

Special morphis ms

Let f be a morphism of a part ially ordered dagger category.

f is monovalued when f ◦ f

†

⊑ 1

Dst f

.

f is entirely deﬁned when f

†

◦ f ⊒ 1

Src f

.

f is injective when f

†

◦ f ⊑ 1

Src f

.

f is s urjective when f ◦ f

†

⊒ 1

Dst f

.

It’s easy to show t hat this is a generalization of monovalued,

entirely deﬁned, injective, a nd surjective b inary relations as

morph isms of the cat egory Rel.

39

Monovalued funcoids

I will denote “atoms a” the set o f at oms under a for an ele-

ment a of a po s et.

The follow ing statements a re equiva lent for a funcoid f:

1. f is mo novalu ed.

2. ∀a ∈ atoms 1

F(Src f )

: hf ia ∈ atoms 1

F(Dst f )

∪

0

F(Dst f )

.

3. ∀I , J ∈ F(Dst f): hf

−1

i(I ⊓ J ) = hf

−1

iI ⊓ hf

−1

iJ .

4. ∀I , J ∈ P(Dst f): hf

−1

i

∗

(I ∩ J) = hf

−1

i

∗

I ⊓ hf

−1

i

∗

J.

Consequently a principal funcoid is mon ovalued i ﬀ its corre-

spondi ng binary relation is monovalued (a fun ct ion).

40

Functions between spaces

Let µ and ν be funcoids corres pond in g to a (pre )topological

spaces , or proximity spaces, or µ and ν be unifo rm spaces

(that is reloids ).

Let f be the funcoid or reloid correspo nding to a function

from the ﬁrst space to the sec o nd spac e.

41

Conti nuous morphis ms

Continui ty, p roximal cont inuity, an d uniform continuity of

f is e xp ressed by the sa m e formu la:

f ◦ µ ⊑ ν ◦ f .

In the case if f is mon ovalued 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 da gger cat-

egories.

42

Relationships of funcoids and rel oids

For every sets A, B we have FCD(A; B) and RLD(A; B)

interrela ted by the below deﬁn ed func ti o ns:

(FCD): RLD(A; B) → FCD(A; B);

(RLD)

in

: FCD(A; B) → RLD(A; B);

(RLD)

o ut

: FCD(A; B) → RLD(A; B).

43

The funcoid induced by a reloid

Every reloid f ∈ RLD (A; B) induces a funcoid (FCD)f ∈

FCD(A; B) by the fo llowing fo rmulas:

X [(FCD)f] Y ⇔ ∀F ∈ GR f : X

↑

FCD(A;B)

F

Y;

h(FCD)f iX =

\

↑

FC D (A;B)

F

X | F ∈ GR f

.

We h ave for every compo s able relo ids f and g:

(FCD)(g ◦ f) = ((FCD)g) ◦ ((FCD)f ).

I will skip some minor fact s on this topic.

44

The reloids induced by a funcoid

Every funco id f ∈ FCD(A; B) induces a reloid from A to B

in two ways, namely intersection o f outward relations and

union of in ward direct products of ﬁlter s:

(R LD)

out

f =

def

l

↑

RLD(A;B)

[GR f];

(RL D)

in

f =

def

G

{A ×

RLD

B | A ∈ F(A), B ∈ F(B), A ×

FC D

B ⊑ f }.

It’s simpl e to show that

(RL D)

in

f =

G

{a ×

RLD

b | a is an ato m of F(A ), b is an atom

of F(A), a ×

FCD

b ⊑ f }.

I will skip some minor resu lts.

45

Some Galoi s connections

For every funcoid f we have

(FCD)(RLD)

in

f = f .

(FCD ): RLD(A; B) → FCD (A; B) is the lower ad joint of

(RL D)

in

: FCD(A; B) → RLD(A; B). Thus

1. (FCD)

F

S =

F

{(FCD)f | f ∈ S };

2. (RLD)

in

d

T =

d

{(RLD)

in

f | f ∈ T }

for every set S of reloids o r T of funcoids.

46

Convergence of funcoids

A ﬁlter F conve rges to a ﬁlt er A regarding to a funcoid

µ (F→

µ

A) iﬀ F ⊑ hµiA. (This generalizes the sta ndard

deﬁnition of ﬁlter convergent to a point or to a set.)

A funcoid f converges to a ﬁlter A regarding to a funcoid µ

(f→

µ

A) iﬀ im f ⊑ hµiA that is iﬀ im f→

µ

A.

A funcoid f converges to a ﬁlter A on a ﬁlter B regarding to

a funcoid µ iﬀ f |

B

→

µ

A.

We can deﬁn e also convergence fo r a reloid f : f→

µ

A⇔im f ⊑

hµiA or what is th e same f→

µ

A ⇔ (FC D)f→

µ

A.

47

Limi t of a funcoid

lim

µ

f = a iﬀ

f→

µ

↑

Dst f

{a}

for a T

2

-sepa rable fu ncoid µ and a n on-empty funcoid f.

It is deﬁned correctly, that i s f h a s no more than one limit.

48

Generalized limit

We can deﬁne a (g en eralized) limit for an arbitrary (dis con-

tinuous ) functi o n, for exam ple any function on the set of

reals, or more generally fro m any topological vector space to

any t o pological vector space, etc.

An idea is th a t t he lim it should not change wh en translating

to a n other point of the spa ce. Thus we need to ﬁx a group

G of translations (or a ny other t ransformations) o f our sp ace.

49