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
