**Algebraic General Topology. Vol 1**:
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Funcoids and Reloids:

a Generalization of Proximities and Uniformities

∗

Victor Porton

September 11, 2013

Abstract

It is a part of my Algebraic General Topology research.

In this article, I introduce the concepts of funcoids, which general-

ize proximity spaces and reloids, which generalize uniform spaces. The

concept of a funcoid is a generalized concept of proximity, the concept of

a reloid is the concept of u niformity cleared (generalized) from superﬂu-

ous details. Also funcoids generalize pretopologies and preclosures. Also

funcoids and reloids are generalizations of binary relations whose domains

and ranges are ﬁlters (instead of sets).

Also funcoids and reloids can be considered as a generalization of (di-

rected) graphs, this provides us a common generalization of analysis and

discrete mathematics.

The concept of continuity is deﬁned by an algebraic formula (instead

of t he old messy epsilon-delta notation) for arbitrary morphisms (includ-

ing funcoids and reloids) of a partially ordered category. In one formula

continuity, proximity continuity, and uniform continuity are generalized.

Contents

1 Common 3

1.1 Earlier works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Used concepts, notation and sta tements . . . . . . . . . . . . . . 3

1.2.1 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Partially ordered dagger categories 5

2.1 Partially ordered categories . . . . . . . . . . . . . . . . . . . . . 5

2.2 Dagger categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Some special classes of morphisms . . . . . . . . . . . . . 7

∗

Keywords: algebraic general topology, quasi-uniform spaces, generalizations of proximity

spaces, generalizations of nearness spaces, generalizations of uniform spaces; A.M.S. subj ect

classiﬁcation: 54J05, 54A05, 54D99, 54E05, 54E15, 54E17, 54E99

1

3 Funcoids 8

3.1 Informal introduction into funcoids . . . . . . . . . . . . . . . . . 8

3.2 Basic deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Composition of funcoids . . . . . . . . . . . . . . . . . . . 11

3.3 Funcoid as continuatio n . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Lattices of funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 More on composition of funcoids . . . . . . . . . . . . . . . . . . 17

3.6 Domain and range of a funco id . . . . . . . . . . . . . . . . . . . 1 9

3.7 Categorie s of funcoids . . . . . . . . . . . . . . . . . . . . . . . . 20

3.8 Spec ifying funcoids by functions o r rela tions on atomic ﬁlter objects 21

3.9 Direct product of ﬁlter objects . . . . . . . . . . . . . . . . . . . 24

3.10 Atomic funco ids . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.11 Complete funcoids . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.12 Completion of funcoids . . . . . . . . . . . . . . . . . . . . . . . . 34

3.13 Monovalued and injective funcoids . . . . . . . . . . . . . . . . . 36

3.14 T

0

-, T

1

- and T

2

-separable funcoids . . . . . . . . . . . . . . . . . 38

3.15 Filter objects closed regarding a funcoid . . . . . . . . . . . . . . 38

4 Reloids 39

4.1 Composition of reloids . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Direct product of ﬁlter objects . . . . . . . . . . . . . . . . . . . 42

4.3 Restricting re loid to a ﬁlter object. Do main a nd image . . . . . . 43

4.4 Categorie s of reloids . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 Monovalued and injective reloids . . . . . . . . . . . . . . . . . . 46

4.6 Complete reloids and completion of reloids . . . . . . . . . . . . . 47

5 Relationships between funcoids and reloids 51

5.1 Funcoid induced by a reloid . . . . . . . . . . . . . . . . . . . . . 51

5.2 Reloids induced by funcoid . . . . . . . . . . . . . . . . . . . . . 56

5.3 Galois connections of funcoids and reloids . . . . . . . . . . . . . 58

6 Continuous morphisms 59

6.1 Tra ditional deﬁnitions of continuity . . . . . . . . . . . . . . . . . 59

6.1.1 Pre-topology . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.1.2 Proximity spaces . . . . . . . . . . . . . . . . . . . . . . . 60

6.1.3 Uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2 Our three deﬁnitions of continuity . . . . . . . . . . . . . . . . . 61

6.3 Continuity of a restricted morphism . . . . . . . . . . . . . . . . 62

7 Connectedness regarding funcoids and reloids 63

7.1 Some lemma s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.2 Endomorphism series . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.3 Connectedness r egarding binary relations . . . . . . . . . . . . . 65

7.4 Connectedness r egarding funcoids and reloids . . . . . . . . . . . 67

7.5 Algebraic properties of S a nd S

∗

. . . . . . . . . . . . . . . . . . 69

2

8 Postface 70

8.1 Misc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A Some counter-examples 70

A.1 Sec ond product. Oblique product . . . . . . . . . . . . . . . . . . 75

1 Common

1.1 Earlier works

Some mathematicians researched generalizations of proximities and uniformities

before me but they have failed to reach the right degre e of generalization which

is presented in this work allowing to represent properties of spaces with algebraic

(or categorical) formulas.

Proximity structures were introduced by Smirnov in [5].

Some references to predecessors:

• In [6], [7], [12], [2], [19] are s tudied generalized uniformities and proximi-

ties.

• Proximities and uniformities are also studied in [10], [11], [18], [20], [21].

• [8] and [9] contains recent progress in quasi-unifor m spaces. [9] has a very

long list of related literature.

Some works ([17]) about proximity spaces consider relations hips of proximities

and compact topological spaces. In this work the attempt to deﬁne or research

their g eneralization, compactness of funcoids or reloids , is not done. It seems

potentially productive to attempt to borrow the deﬁnitions and procedures from

the above mentioned works. I hope to do this study in a separate article.

[4] studies mappings between proximity structures. (In this work no at-

tempt to research mappings between funcoids is done.) [13] researches relation-

ships of quasi-uniform spaces and topological spaces. [1] studies how proximity

structures can be treated as uniform structures and compactiﬁcation regarding

proximity and uniform spaces.

1.2 Used concepts, notation and statements

The set of functions from a set A to a set B is denoted as B

A

.

I will often skip pare ntheses and write f x instead of f (x) to denote the result

of a function f acting on the argument x.

I will call small sets members of some Grothendieck universe. (Let us assume

the axiom of existence of a Grothendieck universe.)

Let f is a sma ll binary relation.

I will denote hf i X = {f α | α ∈ X} and X [f ] Y ⇔ ∃x ∈ X, y ∈ Y : x f y

for small sets X, Y .

3

By just hfi and [f ] I will denote the correspo nding function and relation on

small sets.

λx ∈ D : f (x) = {(x; f (x)) | x ∈ D} for every formula f (x) depended

on a variable x and set D.

I will deno te the least and the greatest element of a poset A as 0

A

and 1

A

respectively.

For elements a and b of a lattice with a minimal element I will denote a ≍ b

when a ∩ b is the minimal element of the lattice and a 6≍ b otherwise. See [15]

for a more gene ral notion.

Propositi on 1 Let f , g, h be binary relations. Then g ◦ f 6≍ h ⇔ g 6≍ h ◦ f

−1

.

Proof

g ◦ f 6≍ h ⇔

∃a, c : a ((g ◦ f ) ∩ h) c ⇔

∃a, c : (a (g ◦ f) c ∧ a h c) ⇔

∃a, b, c : (a f b ∧ b g c ∧ a h c) ⇔

∃b, c :

b g c ∧ b

h ◦ f

−1

c

⇔

∃b, c : b

g ∩

h ◦ f

−1

c ⇔

g 6≍ h ◦ f

−1

.

1.2.1 Filters

In this work the word ﬁlter will refer to a ﬁlter on a set (in contrast to [15]

where ﬁlters are considered on arbitrary posets). Note that I do not require

ﬁlters to be proper.

I will call the set of ﬁlters on a set A (base set) ordere d reverse to set-

theoretic inclusion of ﬁlters the se t of ﬁlter objects on A and denote it F (A)

or just F when the base set is implied a nd call its element ﬁlter objects (f.o.

for short). I will denote up F the ﬁlter corr esponding to a ﬁlter object F. So

we have A ⊆ B ⇔ up A ⊇ up B for every ﬁlter objects A and B on the same set.

In this particular manuscript, we will not equate principal ﬁlter objects with

corres ponding sets as it is done in [15]. Instead we will have Base (A) equal to

the unique base of a f.o. A. I will denote ↑

A

X (or just ↑ X when A is implied)

the principal ﬁlter object on A cor responding to the set X.

Filters are studied in the work [15].

Every set F (A) is a c omplete lattice and we will apply lattice operations to

subsets of such sets without ex plic itly mentioning F (A).

Prior r e ading of [15] is needed to fully understand this work.

Filter objects corresponding to ultraﬁlter s are atoms of the lattice F (A) and

will be called atom ic ﬁlter objects (on A).

4

Also we will need to introduce the concept of generalized ﬁlter base.

Deﬁnition 1 Generalized ﬁlter base is a set S ∈ PF \

0

F

such that

∀A, B ∈ S∃C ∈ S : C ⊆ A ∩ B.

Propositi on 2 Let S is a generalized ﬁlter base. If A

1

, . . . , A

n

∈ S (n ∈

N

),

then

∃C ∈ S : C ⊆ A

1

∩ . . . ∩ A

n

.

Proof C an be easily proved by induction.

Theorem 1 If S is a generalized ﬁlter base, then up

T

S =

S

hupi S.

Proof Obviously up

T

S ⊇

S

hupi S. Reverse ly, let K ∈ up

T

S; then K =

A

1

∩ . . . ∩ A

n

where A

i

∈ up A

i

where A

i

∈ S, i = 1, . . . , n, n ∈

N

; so exists

C ∈ S such that C ⊆ A

1

∩ . . . ∩ A

n

⊆↑ (A

1

∩ . . . ∩ A

n

) =↑ K, K ∈ up C,

K ∈

S

hupi S.

Corollary 1 If S is a generalized ﬁlter base, then

T

S = 0

F

⇔ 0

F

∈ S.

Proof

T

S = 0

F

⇔ ∅ ∈ up

T

S ⇔ ∅ ∈

S

hupi S ⇔ ∃X ∈ S : ∅ ∈ up X ⇔ 0

F

∈

S.

Obvious 1. If S is a ﬁlter base on a set A then

↑

A

S is a generalized ﬁlter

base.

Deﬁnition 2 I will call shifted ﬁltrator a triple (A; Z; ↑) where A and Z are

posets and ↑ is an order embedding from Z to A.

Some concepts and notation can be deﬁned for shifted ﬁltrators through

similar co nc e pts for ﬁltrators : h↑i up a = up

(A;h↑iZ)

a; h↑i Cor a = Cor

(A;h↑iZ)

a,

etc.

For a set A and the set of f.o. F on this set we will consider the shifted

ﬁltrator (F; A; ↑).

2 Partially ordered dagger categories

2.1 Partially ordered categories

Deﬁnition 3 I will call a partially ordered (pre)category a (pre)category

together with partial order ⊆ on each of its Hom-sets with the additional require-

ment that

f

1

⊆ f

2

∧ g

1

⊆ g

2

⇒ g

1

◦ f

1

⊆ g

2

◦ f

2

for every morphisms f

1

, g

1

, f

2

, g

2

such that Src f

1

= Src f

2

∧ Dst f

1

= Dst f

2

=

Src g

1

= Src g

2

∧ Dst g

1

= Dst g

2

.

5

2.2 Dagger categories

Deﬁnition 4 I will call a dagger precategory a precategory together with an

involutive contravariant identity-on-objects prefunctor x 7→ x

†

.

In other words, a dagger precategory is a precategory equipped with a

function x 7→ x

†

on its set of morphisms which reverses the sou rce and the

destination and is subject to the following identities for every morphisms f and

g:

1. f

††

= f;

2. (g ◦ f )

†

= f

†

◦ g

†

.

Deﬁnition 5 I will call a dagger category a category together with an invo-

lutive contravariant identity-on-objects functor x 7→ x

†

.

In other words, a dagger category is a category equipped with a function

x 7→ x

†

on its set of morphisms which reverses the sou rce and the destination

and is subject to the following identities for every morphisms f and g and object

A:

1. f

††

= f;

2. (g ◦ f )

†

= f

†

◦ g

†

;

3. (1

A

)

†

= 1

A

.

Theorem 2 If a category is a dagger precategory then it is a dagger category.

Proof We nee d to prove only that (1

A

)

†

= 1

A

. Really

(1

A

)

†

= (1

A

)

†

◦ 1

A

= (1

A

)

†

◦ (1

A

)

††

= ((1

A

)

†

◦ 1

A

)

†

= (1

A

)

††

= 1

A

.

For a partially ordered dagger (pre)category I will additionally require (for

every morphisms f and g)

f

†

⊆ g

†

⇔ f ⊆ g.

An example of da gger c ategory is the category Rel whose objects are sets and

whose morphisms are binary relations betwe en these sets with usual compositio n

of binary relations and with f

†

= f

−1

.

Deﬁnition 6 A morphism f of a dagger category is called unitary when it is

an isomorphism and f

†

= f

−1

.

Deﬁnition 7 Sy mmetric (endo)morphism of a dagger precategory is such a

morphism f that f = f

†

.

6

Deﬁnition 8 Transitive (endo)morphism of a precategory is such a morphism

f that f = f ◦ f.

Theorem 3 The following conditions are equivalent for a morphism f of a

dagger precategory:

1. f is symmetric and transitive.

2. f = f

†

◦ f.

Proof

(1)⇒(2) If f is symmetric and transitive then f

†

◦ f = f ◦ f = f.

(2)⇒(1) f

†

= (f

†

◦f)

†

= f

†

◦f

††

= f

†

◦f = f, so f is symmetric. f = f

†

◦f =

f ◦ f , s o f is transitive.

2.2.1 Some special classes of morphis ms

Deﬁnition 9 For a partially ordered dagger category I will call monovalued

morphism such a morphism f that f ◦ f

†

⊆ 1

Dst f

.

Deﬁnition 10 For a partially ordered dagger category I will call entirely de-

ﬁned morphism such a morphism f that f

†

◦ f ⊇ 1

Src f

.

Deﬁnition 11 For a partially ordered dagger category I will call injective mor-

phism such a morphism f that f

†

◦ f ⊆ 1

Src f

.

Deﬁnition 12 For a partially ordered dagger category I will call surjective

morphism such a morphism f that f ◦ f

†

⊇ 1

Dst f

.

Remark 1 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.

Obvious 2. “Injective morphism” is a dual of “monovalued morphism” and

“surjective morphism” is a dual of “entirely deﬁned morphism”.

Deﬁnition 13 For a given partially ordered dagger category C the category

of monovalued (entirel y deﬁned, injective, surjective) morphisms of

C is the category with the same set of objects as of C and the set of morphisms

being the set of monovalued (entirely deﬁned, injective, surjective) morphisms

of C with the composition of morphisms the same as in C.

7

We need to prove that these are really categories , that is that composition

of monovalued (entirely deﬁned, injective, surjective) morphisms is monovalued

(entirely deﬁned, injective, surjective) and that identity mor phisms are mo no-

valued, entirely deﬁned, injective, and surjective.

Proof We will prove only for monovalued morphisms and entirely deﬁned

morphisms, a s injective and surjective morphisms are their duals.

Monovalued Let f and g a re monovalued morphisms, Dst f = Src g. (g ◦ f) ◦

(g ◦ f)

†

= g ◦ f ◦ f

†

◦ g

†

⊆ g ◦ 1

Dst f

◦ g

†

= g ◦ 1

Src g

◦ g

†

= g ◦ g

†

⊆ 1

Dst g

=

1

Dst(g◦f )

. So g ◦ f is monovalued.

That identity morphisms are monova lue d follows from the following: 1

A

◦

(1

A

)

†

= 1

A

◦ 1

A

= 1

A

= 1

Dst 1

A

⊆ 1

Dst 1

A

.

Entirely deﬁne d Let f and g are entirely deﬁned morphisms, Dst f = Src g.

(g ◦ f)

†

◦ (g ◦ f) = f

†

◦ g

†

◦ g ◦ f ⊇ f

†

◦ 1

Src g

◦ f = f

†

◦ 1

Dst f

◦ f = f

†

◦ f ⊇

1

Src f

= 1

Src(g◦f )

. So g ◦ f is entirely deﬁned.

That identity morphisms are entirely deﬁned follows from the following:

(1

A

)

†

◦ 1

A

= 1

A

◦ 1

A

= 1

A

= 1

Src 1

A

⊇ 1

Src 1

A

.

Deﬁnition 14 I will call a bijective morphism a morphism which is entirely

deﬁned, monovalued, injective, and surjective.

Obvious 3 . Bijective morphisms form a full subcategory.

Propositi on 3 If a morphism is bijective then it is an isomorphism.

Proof Let f is bijective. Then f ◦ f

†

⊆ 1

Dst f

, f

†

◦ f ⊇ 1

Src f

, f

†

◦ f ⊆ 1

Src f

,

f ◦ f

†

⊇ 1

Dst f

. Thus f ◦ f

†

= 1

Dst f

and f

†

◦ f = 1

Src f

that is f

†

is an inverse

of f.

3 Funcoids

3.1 Informal intro duct ion into funcoids

Funcoids are a generalization of proximity spaces and a generalization of pre-

topologica l spaces. Also funcoids are a generalization of binary relations.

That funcoids are a common generalization of “spaces” (proximity spaces,

(pre)topological spaces) and binary relations (including monovalued functions)

makes them smart for describing properties of functions in regard of spa ces. For

example the statement “f is a continuous function from a space µ to a space

ν” can be described in terms of funcoids as the for mula f ◦ µ ⊆ ν ◦ f (see below

for details).

Most naturally funcoids appear as a generalization of proximity spaces.

8

Let δ be a proximity that is certain binary relation so that A δ B is deﬁned

for every sets A and B. We will extend it from sets to ﬁlter objects by the

formula:

A δ

′

B ⇔ ∀A ∈ up A, B ∈ up B : A δ B.

Then (as it will b e proved below) there exist two functions α, β ∈ F

F

such that

A δ

′

B ⇔ B ∩

F

αA 6= 0

F

⇔ A ∩

F

βB 6= 0

F

.

The pair (α; β) is called funcoid when B ∩

F

αA 6= 0

F

⇔ A ∩

F

βB 6= 0

F

. So

funcoids are a generalization of proximity spaces.

Funcoids consist of two components the ﬁrst α and the s e cond β. The ﬁrst

component of a funcoid f is denoted as hfi and the second component is denoted

as

f

−1

. (The similarity of this notation with the notatio n for the image of a

set under a function is not a coincidence, we will see that in the case of principal

funcoids (see below) these coincide.)

One of the most important properties of a funcoid is that it is uniquely

determined by just one of its components. That is a funcoid f is unique ly

determined by the function hf i. Moreover a funcoid f is uniquely determined

by hf i |

P

S

domhfi

that is by values of function hfi on sets (if we equate principal

ﬁlters with sets).

Next we will consider some examples of funcoids determined by speciﬁed

values of the ﬁrst component o n sets.

Funcoids as a gener alization of pretopological spaces: Let α be a pretopo-

logical space that is a map α ∈ F

℧

for some set ℧. Then we deﬁne α

′

X

def

=

S

F

{αx | x ∈ X} fo r every set X ∈ P ℧. We will prove that there exists a

unique funcoid f such that α

′

= hfi |

P℧

. So funcoids are a generalization of

pretopological spaces. Funcoids ar e also a generalizatio n of preclosure opera-

tors: For every preclosure operator p on a set ℧ it exists a unique funcoid f

such that hf i |

P℧

=↑ ◦p.

For every binary relation p on a set ℧ it exists unique funcoid f such that

∀X ∈ P℧ : hfi ↑ X =↑ hpi X (where hpi is de ﬁned in the introduction), recall

that a funcoid is uniquely determined by the values of its ﬁrst component on

sets. I will call such funcoids principal. So funcoids are a generalization of

binary r elations.

Composition of binary re lations (i.e. of principal funcoids) complies with

the formulas:

hg ◦ fi = hgi ◦ hfi and

(g ◦ f )

−1

=

f

−1

◦

g

−1

.

By the same fo rmulas we can deﬁne comp osition of every two funcoids. Funcoids

with this composition form a category (the category of funcoids).

Also funcoids can be reversed (like reversal of X and Y in a binary relation)

by the formula (α; β)

−1

= (β; α). In particular case if µ is a proximity we have

µ

−1

= µ because proximities are symmetric.

Funcoids behave similarly to (multivalued) functions but acting on ﬁlter

objects instead of acting on sets. Below these will be deﬁned domain and image

of a funcoid (the domain and the image of a funcoid ar e ﬁlter objects).

9

3.2 Basic deﬁnitions

Deﬁnition 15 Let’s call a funcoid from a set A t o a set B a quadruple

(A; B; α; β) where α ∈ F (B)

F(A)

, β ∈ F (A)

F(B)

such that

∀X ∈ F (A) , Y ∈ F (B) : (Y 6≍ αX ⇔ X 6≍ βY).

Further we will assume that all funcoids in consideration are small without

mentioning it explicitly.

Deﬁnition 16 Source and destination of every funcoid (A; B; α; β) are de-

ﬁned as

Src (A; B; α; β) = A and Dst (A; B; α; β) = B.

I will denote FCD (A; B) the set of funcoids from A to B.

I will denote FCD the set of all funcoids (for s mall sets).

Deﬁnition 17 h(A; B; α; β)i

def

= α for a funcoid (A; B; α; β).

Deﬁnition 18 (A; B; α; β)

−1

= (B; A; β; α) for a funcoid (A; B; α; β).

Propositi on 4 If f is a funcoid then f

−1

is also a funcoid.

Proof I t follows from sy mmetry in the deﬁnition of funcoid.

Obvious 4 . (f

−1

)

−1

= f for a funcoid f.

Deﬁnition 19 The relation [f]∈ P (F (Src f) × F (Dst f)) is deﬁned (for every

funcoid f and X ∈ F (Src f), Y ∈ F (Dst f)) by the formula X [f] Y

def

= Y 6≍

hfi X .

Obvious 5. X [f ] Y ⇔ Y 6≍ hfi X ⇔ X 6≍

f

−1

Y for every funcoid f and

X ∈ F (Src f), Y ∈ F (Dst f).

Obvious 6 .

f

−1

=[f]

−1

for a funcoid f .

Theorem 4 Let A, B are small sets.

1. For given value of hfi exists no more than one funcoid f ∈ FCD (A; B).

2. For given value of [f] exists no m ore than one funcoid f ∈ FCD (A; B).

Proof Let f, g ∈ FCD (A; B) .

Obviously hfi = hgi ⇒[f ]=[g] and

f

−1

=

g

−1

⇒[f]=[g]. So it’s enough

to prove that [f]=[g]⇒ hf i = hgi.

Provided that [f ]=[g] we have Y 6≍ hf i X ⇔ X [f ] Y ⇔ X [g] Y ⇔ Y 6≍

hgi X and conseq ue ntly h f i X = hgi X for every X ∈ F (A) and Y ∈ F (B)

because a set of ﬁlter objects is separable [15], thus hfi = hgi.

10

Propositi on 5 hf i 0

F(Src f )

= 0

F(Dst f)

for every funcoid f.

Proof Y 6≍ hf i 0

F(Src f )

⇔ 0

F(Src f )

6≍

f

−1

Y ⇔ 0 ⇔ Y 6≍ 0

F(Dst f)

. Thus

hfi 0

F(Src f)

= 0

F(Dst f )

by separability of ﬁlter objects.

Propositi on 6 hf i (I ∪ J ) = hf i I ∪ hfi J for every funcoid f and I, J ∈

F (Src f).

Proof

⋆ hfi (I ∪ J ) =

{Y ∈ F | Y 6≍ hf i (I ∪ J )} =

Y ∈ F | I ∪ J 6≍

f

−1

Y

= (by corollary 10 in [15])

Y ∈ F | I 6≍

f

−1

Y ∨ J 6≍

f

−1

Y

=

{Y ∈ F | Y 6≍ hfi I ∨ Y 6≍ hf i J } =

{Y ∈ F | Y 6≍ hfi I ∪ hfi J } =

⋆(hfi I ∪ hfi J ).

Thus hf i (I ∪ J ) = hfi I ∪ hfi J because F (Dst f) is separable.

Propositi on 7 For every f ∈ FCD (A; B) for every small sets A and B we

have:

1. K [f ] I ∪ J ⇔ K [f ] I ∨ K [f ] J for every I, J ∈ F (B), K ∈ F (A).

2. I ∪ J [f] K ⇔ I [f ] K ∨ J [f] K for every I, J ∈ F (A), K ∈ F (B).

Proof 1. K [f ] I ∪J ⇔ (I ∪ J )∩hfi K 6= 0

F(B)

⇔ (I ∩ hfi K)∪(J ∩ hfi K) 6=

0

F(B)

⇔ I ∩ hfi K 6= 0

F(B)

∨ J ∩ h f i K 6= 0

F(B)

⇔ K [f ] I ∨ K [f] J .

2. Simila r.

3.2.1 Composition of funcoids

Deﬁnition 20 Fun coids f and g are composable when Dst f = Src g.

Deﬁnition 21 Composition of composable 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

).

Propositi on 8 If f, g are composable fu ncoids then g ◦ f is a funcoid.

Proof Let f = (A; B; α

1

; β

1

), g = (B; C; α

2

; β

2

). For every X ∈ F (A), Y ∈

F (C) we have

Y 6≍ (α

2

◦α

1

)X ⇔ Y 6≍ α

2

α

1

X ⇔ α

1

X 6≍ β

2

Y ⇔ X 6≍ β

1

β

2

Y ⇔ X 6≍ (β

1

◦β

2

)Y.

11

So (A; C; α

2

◦ α

1

; β

1

◦ β

2

) is a funcoid.

Obvious 7 . hg ◦ f i = hgi ◦ hfi for every composable funcoids f and g.

Propositi on 9 (h ◦ g) ◦ f = h ◦ (g ◦ f) for every composable funcoids f , g, h.

Proof

h(h ◦ g) ◦ f i = hh ◦ gi◦hfi = (hhi◦hgi)◦hfi = hhi◦(hgi◦hfi) = hhi◦hg ◦ f i =

hh ◦ (g ◦ f)i.

Theorem 5 (g ◦ f)

−1

= f

−1

◦ g

−1

for every composable fu ncoids f and g.

Proof

(g ◦ f)

−1

=

f

−1

◦

g

−1

=

f

−1

◦ g

−1

.

3.3 Funcoid as c ontinuation

Let f is a funcoid.

Deﬁnition 22 hf i

∗

is the function P (Src f) → F (Dst f) deﬁned by the for-

mula

hfi

∗

X = hfi ↑

Src f

X.

Deﬁnition 23 [f ]

∗

is the relation between P (Src f ) and P (Dst f) deﬁned by

the formula

X [f]

∗

Y =↑

Src f

X [f]↑

Dst f

Y.

Obvious 8 .

1. hfi

∗

= hfi ◦ ↑

Src f

;

2. [f]

∗

=

↑

Dst f

−1

◦ [f] ◦ ↑

Src f

.

Theorem 6 For every funcoid f and X ∈ F (Src f) and Y ∈ F (Dst f)

1. hfi X =

T

hfi

∗

up X ;

2. X [f] Y ⇔ ∀X ∈ up X , Y ∈ up Y : X [f]

∗

Y .

Proof 2. X [f] Y ⇔ Y ∩ hfi X 6= 0

F(Dst f)

⇔ ∀Y ∈ up Y :↑

Dst f

Y ∩ hfi X 6=

0

F(Dst f )

⇔ ∀Y ∈ up Y : X [f ]↑

Dst f

Y .

Analogously X [f ] Y ⇔ ∀X ∈ up X :↑

Src f

X [f ] Y. C ombining these two

equivalences we get

X [f ] Y ⇔ ∀X ∈ up X , Y ∈ up Y :↑

Src f

X [f ]↑

Dst f

Y ⇔ ∀X ∈ up X , Y ∈ up Y : X [f]

∗

Y.

1. Y ∩ hfi X 6= 0

F(Dst f )

⇔ X [f ] Y ⇔ ∀X ∈ up X :↑

Src f

X [f ] Y ⇔ ∀X ∈

up X : Y ∩ hfi

∗

X 6= 0

F(Dst f )

.

12

Let’s denote W =

Y ∩ hfi

∗

X | X ∈ up X

. We will prove that W

is a generalized ﬁlter base. To prove this it is enough to show that V =

hfi

∗

X | X ∈ up X

is a generalized ﬁlter base.

Let P, Q ∈ V . Then P = hf i

∗

A, Q = hfi

∗

B where A, B ∈ up X ; A ∩ B ∈

up X and R ⊆ P ∩ Q for R = hfi

∗

(A ∩ B) ∈ V . So V is a genera lized ﬁlter

base and thus W is a generalized ﬁlter ba se.

0

F(Dst f )

/∈ W ⇔

T

W 6= 0

F(Dst f )

by the corollary 1 of the theorem 1. That

is

∀X ∈ up X : Y ∩ hfi

∗

X 6= 0

F(Dst f )

⇔ Y ∩

\

hfi

∗

up X 6= 0

F(Dst f )

.

Comparing with the a bove, Y ∩ hf i X 6= 0

F(Dst f )

⇔ Y ∩

T

hfi

∗

up X 6=

0

F(Dst f )

. So hfi X =

T

hfi

∗

up X bec ause the lattice of ﬁlter objects is sepa -

rable.

Propositi on 10 For every f ∈ FCD (A; B) we have (for every I, J ∈ PA)

hfi

∗

∅ = 0

F(B)

, hf i

∗

(I ∪ J) = hf i

∗

I ∪ hfi

∗

J

and

¬(I [f ]

∗

∅), I ∪ J [f]

∗

K ⇔ I [f ]

∗

K ∨ J [f ]

∗

K (for every I, J ∈ PA, K ∈ P B),

¬(∅ [f]

∗

I), K [f]

∗

I ∪ J ⇔ K [f ]

∗

I ∨ K [f ]

∗

J (for every I, J ∈ PB, K ∈ P A).

Proof hf i

∗

∅ = hfi ↑

A

∅ = hf i 0

F(A)

= 0

F(B)

; hf i

∗

(I ∪ J) = hfi ↑

A

(I ∪ J) =

hfi

↑

A

I∪ ↑

A

J

= hfi ↑

A

I ∪ hfi ↑

A

J = hfi

∗

I ∪ hfi

∗

J.

I [f ]

∗

∅ ⇔ 0

F(B)

6≍ hfi ↑

A

I ⇔ 0; I ∪ J [f ]

∗

K ⇔↑

A

(I ∪ J) [f ]↑

B

K ⇔↑

B

K 6≍ hfi

∗

(I ∪ J) ⇔↑

B

K 6≍ hfi

∗

I ∪ hfi

∗

J ⇔↑

B

K 6≍ hfi

∗

I∨ ↑

B

K 6≍ hfi

∗

J ⇔

I [f]

∗

K ∨ J [f]

∗

K.

The r est follows from symmetry.

Theorem 7 Fix small sets A and B. Let L

F

= λf ∈ FCD (A; B) : hf i

∗

and

L

R

= λf ∈ FCD (A; B) :[f]

∗

.

1. L

F

is a bijection from the set FCD (A; B) to the set of functions α ∈ F (B)

PA

that obey the conditions (for every I, J ∈ PA)

α∅ = 0

F(B)

, α(I ∪ J) = αI ∪ αJ. (1)

For such α it holds (for every X ∈ F (A))

L

−1

F

α

X =

\

hαi up X . (2)

2. L

R

is a bijection from the set FCD (A; B) to the set of binary relations δ ∈

P (PA × P B) t hat obey the conditions

¬(I δ ∅), I ∪ J δ K ⇔ I δ K ∨ J δ K (for every I, J ∈ PA, K ∈ P B),

¬(∅ δ I), K δ I ∪ J ⇔ K δ I ∨ K δ J (for every I, J ∈ PB, K ∈ P A).

(3)

13

For such δ it holds (for every X ∈ F (A), Y ∈ F (B))

X

L

−1

R

δ

Y ⇔ ∀X ∈ up X, Y ∈ up Y : X δ Y. (4)

Proof Injectivity of L

F

and L

R

, formulas (2) (for α ∈ im L

F

) and (4) (for

δ ∈ im L

R

), formulas (1) and (3) follow from two previous theorems. T he only

thing remained to prove is that for every α and δ that obey the above c onditions

a cor responding funcoid f exists.

2. Let deﬁne α ∈ F (B)

PA

by the formula ∂(αX) = {Y ∈ P B | X δ Y }

for every X ∈ P A. (It is obvious that {Y ∈ P B | X δ Y } is a free star.)

Analogously it can be deﬁned β ∈ F (A)

PB

by the formula ∂(βY ) = {X ∈ PA | X δ Y }.

Let’s continue α and β to α

′

∈ F (B)

F(A)

and β

′

∈ F (A)

F(B)

by the formulas

α

′

X =

\

hαi up X and β

′

Y =

\

hβi up Y

and δ to δ

′

∈ P (F (A) × F (B)) by the formula

X δ

′

Y ⇔ ∀X ∈ up X , Y ∈ up Y : X δ Y.

Y ∩ α

′

X 6= 0

F(B)

⇔ Y ∩

T

hαi up X 6= 0

F(B)

⇔

T

hY∩i hαi up X 6= 0

F(B)

. Let’s

prove that

W = hY∩i hαi up X

is a generalized ﬁlter base: To prove it is enough to show that hαi up X is a

generalized ﬁlter base. If A, B ∈ hαi up X then ex ist X

1

, X

2

∈ up X such that

A = αX

1

and B = αX

2

.

Then α(X

1

∩ X

2

) ∈ hαi up X . So hαi up X is a generalized ﬁlter base and

thus W is a generalized ﬁlter base.

Accordingly to the corolla ry 1 of the theorem 1,

T

hY∩i hαi up X 6= 0

F(B)

is

equivalent to

∀X ∈ up X : Y ∩ αX 6= 0

F(B)

,

what is equivalent to ∀X ∈ up X , Y ∈ up Y :↑

B

Y ∩ αX 6= 0

F(B)

⇔ ∀X ∈

up X , Y ∈ up Y : Y ∈ ∂(αX) ⇔ ∀X ∈ up X , Y ∈ up Y : X δ Y . Combining

the equivalencies we get Y ∩ α

′

X 6= 0

F(B)

⇔ X δ

′

Y. Analogously X ∩ β

′

Y 6=

0

F(A)

⇔ X δ

′

Y. So Y ∩ α

′

X 6= 0

F(B)

⇔ X ∩ β

′

Y 6= 0

F(A)

, that is (A; B; α

′

; β

′

)

is a funcoid. Fr om the fo rmula Y ∩ α

′

X 6= 0

F(B)

⇔ X δ

′

Y it follows that

X [(A; B; α

′

; β

′

)]

∗

Y ⇔↑

B

Y ∩ α

′

↑

A

X 6= 0

F(B)

⇔↑

A

X δ

′

↑

B

Y ⇔ X δ Y.

1. Let deﬁne the relation δ ∈ P (P A × PB) by the fo rmula X δ Y ⇔↑

B

Y ∩ αX 6= 0

F(B)

.

That ¬(∅ δ I) and ¬(I δ ∅ ) is obvious. We have I∪J δ K ⇔↑

B

K∩α(I∪J) 6=

0

F(B)

⇔↑

B

K ∩(αI ∪ αJ) 6= 0

F(B)

⇔↑

B

K ∩αI 6= 0

F(B)

∨ ↑

B

K ∩αI 6= 0

F(B)

⇔

I δ K ∨ J δ K and

K δ I ∪ J ⇔↑

B

(I ∪ J) ∩ αK 6= 0

F(B)

⇔

↑

B

I∪ ↑

B

J

∩ αK 6= 0

F(B)

⇔

↑

B

I ∩ αK

∪

↑

B

J ∩ αK

6= 0

F(B)

⇔↑

B

I ∩ αK 6= 0

F(B)

∨ ↑

B

J ∩ αK 6=

0

F(B)

⇔ K δ I ∨ K δ J.

14

That is the formulas (3) are true.

Accordingly the above there exists a funcoid f such that

X [f] Y ⇔ ∀X ∈ up X , Y ∈ up Y : X δ Y.

∀X ∈ P A, Y ∈ PB :

↑

B

Y ∩ hf i ↑

A

X 6= 0

F(B)

⇔↑

A

X [f ]↑

B

Y ⇔ X δ Y ⇔↑

B

Y ∩ αX 6= 0

F(B)

,

consequently ∀X ∈ P A : αX = hf i ↑

A

X = hfi

∗

X.

Note that by the last theorem to every proximity δ corresponds a unique

funcoid. So funcoids are a generaliz ation of (quasi-)proximity structures.

Reverse funcoids c an be cons ide red as a generalization of conjugate quasi-

proximity.

Deﬁnition 24 Any small (multivalued) function F : A → B corresponds to

a funcoid ↑

FCD(A;B)

F ∈ FCD (A; B), where by deﬁnition

↑

FCD(A;B)

F

X =

T

↑

B

hhF ii up X for every X ∈ F (A).

Using the last theorem it is easy to show that this deﬁnition is monovalued

and does not contradict to former stuﬀ. (Take α =↑

B

◦ hF i.)

Deﬁnition 25 Fun coids corresponding to a binary relation (= multivalued func-

tion) are called principal funcoids.

We may eq uate principal funcoids with cor responding binar y r elations by

the method of appendix B in [15]. This is useful for de scribing r elationships o f

funcoids and binary relations, such as for the formulas of continuous functions

and continuous funcoids (see below).

Theorem 8 If S is a generalized ﬁlter base on Sr c f then hfi

T

S =

T

hhfii S

for every funcoid f.

Proof hf i

T

S ⊆ hf i X for e very X ∈ S and thus hfi

T

S ⊆

T

hhfii S.

By properties of generaliz ed ﬁlter bases:

hfi

T

S =

T

hfi

∗

up

T

S =

T

hfi

∗

{X | ∃P ∈ S : X ∈ up P} =

T

hfi

∗

X | ∃P ∈ S : X ∈ up P

⊇

T

{hfi P | P ∈ S} =

T

hhfii S.

3.4 Lattices of funcoids

Deﬁnition 26 f ⊆ g

def

= [f]⊆[g] for f, g ∈ FCD.

Thus every FCD (A; B) is a po set. (It’s taken into account that [f ]6=[g] if

f 6= g<