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Cauchy Filters on R eloi ds

by Victor Porton

Email: porton@narod.ru

Web: http://www.mathematics21.org

March 7, 2014

Abstract

In this article I consider low ﬁlters on reloids, generalizing Cauchy ﬁlters on uniform spaces.

Using low ﬁlters, I deﬁne Cauchy-complete reloids, generalizing complete uniform spaces.

1 Preface

This is a preliminary partial draft.

To understand this article you need ﬁrst look into my book [1].

As my book is yet in preprint stage and I may change it, I probably will integrate the content

of this article into the book.

http://math.stackexchange.com/questions/401989/what-are-interesting-properties-of- totally-

bounded-uniform-spaces

http://ncatlab.org/nlab/show/proximity+space#uniform_spaces for a proof sketch that prox-

imities correspond to totally bounded uniformi ties.

2 Low ﬁlters space

Deﬁnition 1. A lower set

1

of proper ﬁlters on U (a set) is a set C of proper ﬁlters on U, such

that if 0

G ⊑ F and F ∈ C then G ∈ C . [TODO: Probably should include the improper ﬁlter.]

Deﬁnition 2. I call low ﬁlters space a set together with a lower set of proper ﬁlters on this set.

Deﬁnition 3. PR(U; C ) = C ; Ob(U ; C ) = U .

2

Deﬁnition 4. Introduce an order on low ﬁlters spaces: (U; C ) ⊑ (U ; D ) ⇔ C ⊑ D .

3 Cauchy spaces

Deﬁnition 5. A Cauchy space on a set X is a low ﬁlters space (U; C ) (element of C are called

Cauchy ﬁlters) such that:

1. ∀x ∈ U : ↑

X

{x} ∈ C ;

2. If F, G are Cauchy ﬁlters and F

G then F ⊔ G is a Cauchy ﬁlter.

Deﬁnition 6. A completely Cauchy space on a set X is a low ﬁlters space (U; C ) (element of C

are called Cauchy ﬁlters) such that:

1. ∀x ∈ X: ↑

X

{x} ∈ C ;

2. If S is a nonempty set of Cauchy ﬁlters and

d

S

0

F(X)

then

F

S is a Cauchy ﬁlter.

1. Remember that our orders on ﬁlters is the reverse to set theoretic inclusion. It could be called an upper set

in other sources.

2. PR is from English word proﬁle.

1

Obvi ous 7. Every completely Cauchy space is a Cauchy space.

Proposition 8.

F

{X ∈C | X ⊒F }

S =

F

S for nonempty S ∈ P {X ∈ C | X ⊒ F }, provided that F

is a ﬁxed Cauchy ﬁlter on a completely Cauchy space.

Proof. F is proper. So for every nonempty S ∈ P {X ∈ C | X ⊒ F } we have

d

S ⊒ F

0

F(X)

.

Thus

F

S is a Cauchy ﬁlter and so

F

S ∈ {X ∈ C | X ⊒ F }.

Proposition 9. If F is a ﬁxed Cauchy ﬁlte r on a completely Cauchy space, then the poset

{X ∈ C | X ⊒ F } (wit h th e induced order) is a complete lattice.

Proof. If S

∅ then

F

{X ∈C | X ⊒F }

S =

F

S. If S = ∅ then

F

{X ∈C | X ⊒F }

S = F.

Corollary 10. If F is a ﬁxed Cauchy ﬁlter on a completely Cauchy space, then the poset {X ∈

C | X ⊒ F } (with the induced order) has a maximum.

4 Relationships with s ymmetri c reloids

Deﬁnition 11. Denote (RLD)

Low

(U ; C ) =

F

{X ×

RLD

X | X ∈ C }.

Deﬁnition 12. (Low)ν (low ﬁlters for reloid ν) is a l ow ﬁlters space on U such that

PR (Low)ν = {X ∈ F

U

\ {0

F

} | X ×

RLD

X ⊑ ν }.

Theorem 13. If (U ; C ) is a low ﬁlters space, then (U ; C ) = (Low)(RLD)

Low

(U ; C ).

Proof. If X ∈ C then X ×

RLD

X ⊑ (RLD)

Low

(U; C ) a nd thus X ∈ PR (Low)(RLD)

Low

(U; C ). Thus

(U ; C ) ⊑ (Low)(RLD)

Low

(U; C ).

Let’s prove (U; C ) ⊒ (Low)(RLD)

Low

(U; C ).

Let A ∈ PR (Low)(RLD)

Low

(U ; C ). We need to prove A ∈ C .

Really A ×

RLD

A ⊑ (RLD)

Low

(U; C ). It is enough to prove that ∃X ∈ C : A ⊑ X .

Suppose ∄X ∈ C : A ⊑ X .

For every X ∈ C obtain X

X

∈ X such that X

X

A (if forall X ∈ X we have X

X

∈ A, then X ⊒ A

what is contrary to our supposition).

It is now enough to prove A ×

RLD

A ⊑

F

{↑

U

X

X

×

RLD

↑

U

X

X

| X ∈ C }.

Really,

F

{↑

U

X

X

×

RLD

↑

U

X

X

| X ∈ C } = ↑

RLD(U ;U)

S

{X

X

× X

X

| X ∈ C }. So our claim takes

the form

S

{X

X

× X

X

| X ∈ C }

GR(A ×

RLD

A) that is ∀A ∈ A:

S

{X

X

× X

X

| X ∈ C } + A × A

what is true be cause X

X

+ A for ever y A ∈ A.

Remark 14. The last theorem does not hold with X ×

FCD

X instead of X ×

RLD

X (take C =

{{x} | x ∈ U } for an inﬁnite set U as a counter-example).

Remark 15. Not every symmetric reloid is in the form (RLD)

Low

(U; C ) for some Cauchy space

(U ; C ). The same Cauchy space can be induced by diﬀerent uniform spaces. Se e http://math.stack-

exchange.com/questions/702182/diﬀerent-uniform-spaces-having-the-same-set-of-cauchy-ﬁlters

[TODO: Is composition of two images of low ﬁlter spaces also a low ﬁlters space?]

5 More on Cauchy ﬁlt ers

Obvi ous 16. Low ﬁlter on an endoreloid ν is a ﬁlter F such that

∀U ∈ GR f ∃A ∈ F: A × A ⊆ U .

Remark 17. The above formula is the standard deﬁnition of Cauchy ﬁlters on uniform spaces.

2 Section 5

Proposition 18. If ν ⊒ ν ◦ ν

−1

then every neighborhood ﬁlter is a Cauchy ﬁlter, that it

ν ⊒ h(FCD)ν i

∗

{x} ×

RLD

h(FCD)νi

∗

{x}

for every point x.

Proof. h(FCD)ν i

∗

{x} ×

RLD

h(FCD)νi

∗

{x} = h(FCD)ν i↑

Ob ν

{x} ×

RLD

h(FCD)νi↑

Ob ν

{x} = ν ◦

(↑

Ob ν

{x} ×

RLD

↑

Ob ν

{x}) ◦ ν

−1

= ν ◦

↑

RLD(Ob ν;Ob ν)

{(x; x)}

◦ ν

−1

⊑ ν ◦ id

RLD(Ob ν;Ob ν)

◦ ν

−1

=

ν ◦ ν

−1

⊑ ν.

Proposition 19. If a ﬁlte r converges to a point, it is a low ﬁlter, provided that every neighborhood

ﬁlter is a low ﬁlter.

Proof. Let F ⊑ h(FCD)ν i

∗

{x}. T hen F ×

RLD

F ⊑ h(FCD)ν i

∗

{x} ×

RLD

h(FCD)νi

∗

{x} ⊑ ν.

Corollary 20. If a ﬁlter converges to a po int, it is a low ﬁlter, provided that ν ⊒ ν ◦ ν

−1

.

6 Maximal Cauchy ﬁlters

Lemma 21. Let S be a set of sets with

d

h↑

F

iS

0

F

(in other words, S has ﬁnite intersection

property). Let T = {X × X | X ∈ S }. Then

[

T ◦

[

T =

[

S ×

[

S.

Proof. Let x ∈

S

S. Then x ∈ X for some X ∈ S. h

S

T i{x} ⊒ ↑X ⊇

T

S

∅. Thus

h

S

T ◦

S

T i{x} = h

S

T ih

S

T i{x} ∈ h↑

FCD

S

T i

d

h↑

F

iS ⊒

F

{h↑

FCD

(X × X)i

d

h↑

F

iS | X ∈

S } =

F

{↑

F

X | X ∈ S } =

F

h↑

F

iS that is h

S

T ◦

S

T i{x} ⊇

S

S.

Corollary 22. Let S be a set of ﬁlters (on some ﬁxed set) with nonempty meet. Let

T = {X ×

RLD

X | X ∈ S }

Then

G

T ◦

G

T =

G

S ×

RLD

G

S.

Proof.

F

T ◦

F

T =

d

{↑

F

(X ◦ X) | X ∈

F

T }.

If X ∈

F

T then X =

S

Q∈T

(P

Q

× P

Q

) where P

Q

∈ Q. Therefore by the lemma we have

[

{P

Q

× P

Q

| Q ∈ T } ◦

[

{P

Q

× P

Q

| Q ∈ T } =

[

Q∈T

P

Q

×

[

Q∈T

P

Q

.

Thus X ◦ X =

S

Q∈T

P

Q

×

S

Q∈T

P

Q

.

Consequently

F

T ◦

F

T =

d

↑

F

S

Q∈T

P

Q

×

S

Q∈T

P

Q

| X ∈

F

T

⊒

F

S ×

RLD

F

S.

F

T ◦

F

T ⊑

F

S ×

RLD

F

S is obvious.

Deﬁnition 23. I call an endorelo id ν symmetrically tra nsitive iﬀ for every symmetric endofuncoid

f ∈ FCD(Ob ν; Ob ν) we have f ⊑ ν ⇒ f ◦ f ⊑ ν.

Obvi ous 24. It is symmetrically transitive if at le ast one of the following holds:

1. ν ◦ ν ⊑ ν;

2. ν ◦ ν

−1

⊑ ν;

3. ν

−1

◦ ν ⊑ ν.

4. ν

−1

◦ ν

−1

⊑ ν.

Corollary 25. Every uniform space is symmetrically transitive.

Maximal Cauchy filters 3

Proposition 26. (Low)ν is a completely Ca uchy space for every symmetrically transitive

endoreloid ν.

Proof. Suppose S ∈ P {X ∈ F \ {0

F

} | X ×

RLD

X ⊑ ν } and S

∅.

F

{X ×

RLD

X | X ∈ S} ⊑ ν;

F

{X ×

RLD

X | X ∈ S} ◦

F

{X ×

RLD

X | X ∈ S} ⊑ ν;

F

S ×

RLD

F

S ⊑ ν (taken into account that S has nonempty meet). Thus

F

S is Cauchy.

Proposition 27. The neighbourhood ﬁlter h(FCD)ν i

∗

{x} of a point x ∈ Ob ν is a maximal Cauchy

ﬁlter, if it is a Cauchy ﬁlter and ν is a reﬂexive reloid.

[TODO: Does it holds for all low ﬁlters?]

Proof. Let N = h(FCD)ν i

∗

{x}. Let C ⊒ N be a Cauchy ﬁlter. We need to s how N ⊒ C .

Since C is Cauchy ﬁlter, C ×

RLD

C ⊑ν. Since C ⊒ N we have C is a neighborhood of x and thus

↑

Ob ν

{x}⊑ C (reﬂexivity of ν). Thus ↑

Ob ν

{x}×

RLD

C ⊑ C ×

RLD

C and hence ↑

Ob ν

{x} ×

RLD

C ⊑ ν;

C ⊑ im(ν |

↑

Ob ν

{x}

) = h(FCD)ν i

∗

{x} = N .

7 Cauchy continuous functions

Deﬁnition 2 8. A function f : U → V is Cauchy continuous from a low ﬁlters space (U; C ) to a

low ﬁlters space (V ; D) when ∀X ∈ C : h↑

FCD

f iX ∈ D .

Proposition 29. Let f is a pr incipal reloid. Then f ∈ C((RLD)

Low

C ; (R LD )

Low

D) iﬀ f is Cauchy

continuous.

f ◦ (RLD)

Low

C ◦ f

−1

⊑ (RLD)

Low

D ⇔

G

{f ◦ (X ×

RLD

X ) ◦ f

−1

| X ∈ C } ⊑ (RLD)

Low

D ⇔

G

{h↑

FCD

f iX ×

RLD

h↑

FCD

f iX | X ∈ C } ⊑ (RLD)

Low

D ⇔

∀X ∈ C : h↑

FCD

f iX ×

RLD

h↑

FCD

f iX ⊑ (RLD)

Low

D ⇔

∀X ∈ C : h↑

FCD

f iX ∈ D .

Thus we have expres sed Cauchy properties through the algebra of reloids.

8 Cauchy-complete relo ids

Deﬁnition 30. An endoreloid ν is Ca uchy-complete iﬀ every low ﬁlter for this reloid converges to

a point.

Remark 31. In my book [1] comple te reloid means something diﬀerent. I will always prepend the

word “Cauchy” to the word “complete” whe n meaning is by the last deﬁnition.

https://en.wikipedia.org/wiki/Complete_uniform_space#Completeness

9 Totally bounded

http://ncatlab.org/nlab/show/Cauchy+space

Deﬁnition 32. Cauchy space is called totally bounded when every proper ﬁlter contains a Cauchy

ﬁlter.

Obvi ous 33. A re loid ν is to tally bounded iﬀ

∀X ∈ P Ob ν ∃X ∈ F

Ob ν

: (0

X ⊑ ↑

Ob ν

X ∧ X ×

RLD

X ⊑ ν).

4 Section 9

Theorem 34. A symmetric transitive reloid is totally bounded iﬀ its Cauchy space is totally

bounded.

Proof.

⇒. Let F be a proper ﬁlter on Ob ν and let a ∈ atoms F. It’s enough to prove that a is Cauchy.

Let D ∈ GR ν. Let also E ∈ GR ν is symmetric and E ◦ E ⊆ D. There existsa ﬁnite subset

F ⊆ Ob ν such that hE iF = Ob ν. Then obvio usly exists x ∈ F such that a ⊑ ↑

Ob ν

hE i{x},

but hE i{x} × hE i{x} = E

−1

◦ ({x} × {x}) ◦ E ⊆ D, thus a ×

RLD

a ⊑ ↑

RLD(Ob ν;Ob ν)

D.

Because D was taken arbit rary, we have a ×

RLD

a ⊑ ν that is a is Cauchy.

⇐. Suppose that Cauchy space associated with a reloid ν is totally bounded but the reloid

ν isn’t totally bounded. So the re exists a D ∈ GR ν such that (Ob ν) \ hDiF

∅ for every

ﬁnite set F .

Consider the ﬁlter base

S = {(Ob ν) \ hD iF | F ∈ P Ob ν , F is ﬁnite}

and the ﬁlter F =

d

h↑

Ob ν

iS generated by this base. The ﬁlter F is proper be cause

intersection P ∩ Q ∈ S for every P , Q ∈ S and ∅

S. Thus there exists a Cauchy (for our

Cauchy space) ﬁlter X ⊑ F that is X ×

RLD

X ⊑ ν.

Thus there exists M ∈ X such that M × M ⊆ D . Let F be a ﬁnite subset of Ob ν.

Then (Ob ν) \ hDiF ∈ F ⊒ X . Thus M

(Ob ν) \ hDiF and so there exists a point

x ∈ M ∩ ((Ob ν) \ hDiF ).

hM × M i{p} ⊆ hDi{x} for every p ∈ M; thus M ⊆ hDi{x}.

So M ⊆ hD i(F ∪ {x}). But this means that M ∈ X does not intersect (Ob ν) \

hDi(F ∪ {x}) ∈ F ⊒ X , what is a contradiction (taken into account that X is proper).

http://math.stackexchange.com/questions/104696/pre-compactness-total-boundedness-and-

cauchy-sequential-compactness

10 Totally bounded funcoids

Deﬁnition 35. A funcoid ν is totally bounde d iﬀ

∀X ∈ Ob ν ∃X ∈ F

Ob ν

: (0

X ⊑ ↑

Ob ν

X ∧ X ×

FCD

X ⊑ ν).

This can be rewritten in elementary terms (without using fu nc oidal product:

X ×

FCD

X ⊑ ν ⇔ ∀P ∈ ∂X : X ⊑ hν iP ⇔ ∀P ∈ ∂X , Q ∈ ∂X : P [ν]

∗

Q ⇔ ∀P , Q ∈ Ob ν:

(∀E ∈ X : (E ∩ P

∅ ∧ E ∩ Q

∅) ⇒ P [ν]

∗

Q).

Note that probably I am the ﬁrst p e rson which has written the above formula (for p roximity

spaces for instance) explicitly.

11 On principal low ﬁ lter spaces

Deﬁnition 36. A low ﬁlter sp ace (U ; C ) is principal when all ﬁlters in C are principal.

Deﬁnition 37. A low ﬁlter sp ace (U ; C ) is reﬂexive when ∀x ∈ U : ↑

U

{x} ∈ C .

Proposition 38. Having ﬁxed a set U , principal reﬂexive low ﬁlter spaces on U bijectively cor-

respond to principal reﬂexive symmertic endoreloids on U.

Proof. ??

http://math.stackexchange.com/questions/701684/union-of-cartesian-squares

On principal low filter spaces 5

12 Rest

https://en.wikipedia.org/wiki/Cauchy_ﬁlter#Cauchy_ ﬁlters

https://en.wikipedia.org/wiki/Uniform_space “Hausdorﬀ completion of a uniform space” here)

http://at.yorku.ca/z/a/a/b/13.htm : the category Prox of proximity spaces and proximally

continuous maps (i.e. maps preserving nearness between two sets) is isomorphic to the category

of totally bounded uniform spaces (and uniformly continuous maps).

https://en.wikipedia.org/wiki/Cauchy_space http://ncatlab.org/nlab/show/Cauchy+space

http://arxiv.org/abs/1309.1748

http://projecteuclid.org/download/pdf_1/euclid.pja/1195521991

http://www.emis.de/journals/HOA/IJMMS/Volu me5_3/404620.pdf

~/math/books/Cauchy_spaces.pdf

Bib liography

[1] Victor Porton. Algebraic General Topology. Volume 1. 2013.

6 Section