GMaynezMancioAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 8, No. 2, 2007

pp. 213-221

Completions of pre–uniform spaces

Adalberto Garćıa–Máynez and Rubén S. Mancio T.

Abstract. In this paper we prove the existence of a completion
of a T0–pre-uniform space (X, U), with the property that each Cauchy
filter in (X, U) contains a weakly round filter.

2000 AMS Classification: 54A05, 54A20, 54E15.

Keywords: pre–uniform space, complete pre–uniform space, uniform conti-
nuity, pre–uniformity basis, semi–uniformity basis, unimorphism, unimorphism
embedding, equivalent, admisible and compatible pre–uniformities, balanced
filter, Cauchy, minimal Cauchy, round, weakly round, and strongly round fil-
ters, proper pre–uniform space.

1. Introduction

By a pre–uniform space we mean a pair (X, U) where X is a set and U is
a non–empty family of covers of X satisfying certain properties. Every pre-
uniform space (X, U) determines a topology TU in X and the convergence prop-
erties of filters in (X, TU ) constitute an important area of study. Pre–uniform
spaces generalize the semi–uniform spaces introduced by Morita in [3]. The
most important filters to be considered are Cauchy filters in (X, U), i.e., filters
in X which intersect every cover of U. In many important examples, for a
Cauchy filter η in (X, U) to converge it is necessary and sufficient that η has
an adherence point, i.e., a point which belongs to the closure of every member
of η.

We consider an increasing chain of four important subclasses of the class
of Cauchy filters: strongly round filters, round filters, weakly round filters
and minimal filters. Every Cauchy filter in a semi–uniform space contains an
strongly round filter and hence, a minimal filter. However, this is not true in
more general pre–uniform spaces.

We give all the necessary definitions in next section.
As in the case of uniform or semi–uniform spaces, we say that a pre–uniform

space (X, U) is complete if every Cauchy filter in (X, U) converges. We have



214 A. Garćıa–Máynez and R. Mancio

four less restrictive types of completeness if we require only that the filters in
one of the four classes defined above are convergent.

If (X, U) is a pre–uniform space and A ⊆ X, the family of cover restrictions
UA =

{
α|A : α ∈ U

}
determines a pre–uniform space (A, UA) and we say then

that (A, UA) is a subspace of (X, U) or that (X, U) is an extension of (A, UA). It
is easy to see that TU |A = TUA and this justifies the use of the words “subspace”
and “extension”.

In this paper we prove that every T0–pre–uniform space (X, U) has a canon-

ical T1 extension (X̂, Û ), where every weakly round filter converges. We also

prove that a necessary and sufficient condition for (X̂, Û ) to be complete is that
every Cauchy filter in (X, U) contains a weakly round filter. As a corollary of
this, we deduce the known result that every semi–uniform space is completable.
These are the main results of this paper.

2. Pre–uniform spaces and uniform continuity

All the definitions and notation agreements of this paper appear in the
doctoral thesis of the second author. For convenience to the reader, we shall
include the most important ones.

Note 1. If X is a set, F ⊆ X, p ∈ X and α is a cover of X, then:

S T (p, α) = S T ({p} , α) = ∪ {L ∈ α : p ∈ L}

S T (F, α) = ∪ {L ∈ α : L ∩ F 6= ∅} .

Definition 2.1. Let U be a non–empty family of covers of a set X. The topology
TU in X induced by U is defined as follows:

L ⊆ X belongs to TU iff for every x ∈ L, we may find a finite

collection {α1, α2, . . . , αn} ⊆ U such that
n⋂

i=1

S T (x, αi) ⊆ L.

Note 2. Suppose α1, α2, . . . , αn, α, β are covers of a set X. We denote:

α1 ∧ α ∧ · · · ∧ αn = {L1 ∩ L2 ∩ · · · ∩ Ln : L1 ∈ α1, L2 ∈ α2, · · · , Ln ∈ αn} .

α ≤ β means that α refines β i.e., there exists a map λ : α → β such that
A ⊆ λ (A) for every A ∈ α. Clearly α1 ∧ α2 ∧ · · · ∧ αn ≤ αi for every i ∈
{1, 2, . . . , n}.

If U1, U2 are collections of covers of a set X, we write U1 ≤ U2 if for every
α ∈ U1, there exists a finite collection {β1, β2, . . . , βn} ⊆ U2 such that β1 ∧
β2 ∧ · · · ∧ βn ≤ α. U1 and U2 are equivalent if U1 ≤ U2 and U2 ≤ U1.

Clearly U1 ⊆ U2 ⇒ U1 ≤ U2 ⇒ TU1 ⊆ TU2 .
If U is a collection of covers a set X, U+ denotes the family of covers γ of X

such that for some α ∈ U, we have α ≤ γ. Clearly, the families U and U+ are
equivalent and hence they generate the same topology on X.



Completions of pre–uniform spaces 215

Definition 2.2. A non-empty collection U of covers of a set X is a pre–
uniformity basis on X if U satisfies the following two conditions:

1) Whenever α1, α2, . . . , αn ∈ U there exists a cover β ∈ U such that
β ≤ α1 ∧ α2 ∧ · · · ∧ αn.

2) For each α ∈ U, there exists a cover β ∈ U such that β ≤
◦
α, where

◦
α = {int TU L : L ∈ α}.

Hence, every pre–uniformity basis U is equivalent to a pre–uniformity basis
U′ where each cover α ∈ U′ is open with respect to the topology TU .

Definition 2.3. A pre–uniform space is a pair (X, U), where X is a set and U
is a pre–uniformity basis on X.

Remark 2.4. If (X, U) is a pre–uniform space and A ⊆ X, then U |A ={
α|A : α ∈ U

}
is a pre–uniformity basis on A and the topologies TU |A and

TU |A coincide. We say then that
(
A, U |A

)
is a pre–uniform subspace of (X, U).

Remark 2.5. Let (X, U) be a pre–uniform space. Then, for every A ⊆ X, we
have:

(2.1) intTU A = {x ∈ A : there exists αx ∈ U such that S T (x, αx) ⊆ A} .

Definition 2.6.

1) A map f : (X, U) → (Y, V) between pre–uniform spaces is uniformly
continuous if for every β ∈ V, we may find a cover α ∈ U such that
α ≤

{
f −1 (B) : B ∈ β

}
.

2) A bijection ϕ : (X, U) → (Y, V) between pre–uniform spaces is said to
be a unimorphism if ϕ and ϕ−1 are both uniformly continuous maps
and we say, in this case, that (X, U) and (Y, V) are unimorphic spaces.

3) A map g : (X, U) → (Y, V) from the pre–uniform space (X, U) into the
pre–uniform space (Y, V) is a unimorphic embedding if g (X) is dense

in Y and g is a unimorphism from (X, U) onto
(
g (X) , V |g (X)

)
.

Remark 2.7.

1) If f : (X, U) → (Y, V) is uniformly continuous, then f : (X, TU ) →
(Y, TV ) is continuous.

2) Si U, V are pre–uniformity bases in the same set X, then the identity
map id : (X, U) → (X, V) is uniformly continuous iff V ≤ U. Therefore,
id is a unimorphism iff U and V are equivalent.



216 A. Garćıa–Máynez and R. Mancio

Definition 2.8 (See [2], Definition 1.2.2, page 10).

1) A non–empty family U of covers of a set X is a semi–uniformity basis
on X if U satisfies 2.2.1 and:
SU) For every α ∈ U, there exists a cover β ∈ U such that for every

B ∈ β, we may find a cover γ B ∈ U and a set A B ∈ α such that
S T (B, γ B) ⊆ A B.

2) A semi–uniform space is a pair (X, U), where X is a set and U is a
semi–uniformity basis on X.

3) U is a uniformity basis on X if U satisfies condition 2.2.1 and the
stronger condition:
U) For every α ∈ U, there exists a cover β ∈ U such that:

{ST (B, β) : B ∈ β} ≤ α.

The following facts are well known:

Theorem 2.9.

1) Let U be a semi–uniformity basis on a set X. Then TU is a a regular
topology on X. Conversely, if (X, T ) is a regular topological space, there
exists a semi–uniformity basis U on X such that T = TU .

2) Let U be a uniformity basis on a set X. Then TU is a a completely
regular topology on X. Conversely, if (X, T ) is a completely regular
topological space, there exists a uniformity basis U on X such that T =
TU .

3. Filters and completeness

We recall now some definitions about filters.
Let X be a set. A filter in X is a non–empty subfamily F of the power set

P (X) which satisfies the following properties:

i) ∅ /∈ F.
ii) F1, F2 ∈ F ⇒ F1 ∩ F2 ∈ F.

iii) F ∈ F, F ⊆ L ⊆ X ⇒ L ∈ F.

A filter base in X is a non–empty subfamily η of P (X) satisfying the prop-
erties:

i) ∅ /∈ η.
ii) N1, N2 ∈ η ⇒ ∃ N3 ∈ η such that N3 ⊆ N1 ∩ N2.

For any subfamily G ⊆ P (X), we denote:

G+ = {A ∈ P (X) : G ⊆ A for some G ∈ G} .

If G1, G2 ⊆ P (X), G1 ≤ G2 means that G
+
2 ⊆ G

+
1 , i.e., G1 ≤ G2 iff every element

of G2 contains an element of G1. G1 ∼ G2 (G1 is equivalent to G2) means that

G1 ≤ G2 and G2 ≤ G1, i.e., G1 ∼ G2 iff G
+
1 = G

+
2 . This is clearly an equivalence

relation in P (P (X)). If we restrict ourselves to filter basis in X, ∼ is still an
equivalence relation. It is easy to prove that every equivalence class contains



Completions of pre–uniform spaces 217

exactly one filter, namely, if η is a filter base in X, η+ is the only filter in X
which satisfies the relation η ∼ η+.

Let (X, T ) be a topological space and let η be a filter in X. A point x ∈ X
is an adherence point of η (in symbols, η 7→ x) if every neighborhood of x
intersects every element of η. Equivalently, η 7→ x iff x ∈ Cℓ (N) (= the T –
closure of N) for every N ∈ η. x is a convergence point of η (in symbols, η → x)
if every neighborhood of x belongs to η.

A filter η in a set X is an ultrafilter in X if η is not properly contained in
any other filter in X. Two filters η1, η2 in X mingle (in symbols, η1 ↔ η2) if
every element of η1 intersects every element of η2.

Remark 3.1. Two filters η1, η2 in X mingle iff there exists a filter η in X such
that η1 ∪ η2 ⊆ η.

A filter η in a topological space (X, T ) is balanced if every adherence point
of η is also a convergence point of η. For any x ∈ X, ηx denotes the filter of
T –neighborhoods of x.

Remark 3.2. Let η1, η2 be filters in a topological space (X, T ) and let x ∈ X.
Then:

1) η1 → x implies η1 7→ x.
2) η1 → x and η1 ↔ η2 imply that η2 7→ x.
3) η1 7→ x iff η1 ↔ ηx.
4) η1 → x iff η1 ⊇ ηx.
5) η1 ≤ η2 and η1 7→ x imply that η2 7→ x.
6) η1 ≤ η2 and η2 → x imply that η1 → x.
7) Every ultrafilter in X is balanced.
8) Every filter in X without adherence points is balanced.
9) Every convergent filter in a Hausdorff space is balanced.

Definition 3.3. A filter η in a pre–uniform space (X, U) is U–Cauchy (or
Cauchy in (X, U)) if for every α ∈ U, we have η ∩ α 6= ∅.

Remark 3.4. If ϕ : (X, U) → (Y, V) is a uniformly continuous map and if

F is a U–Cauchy filter, then ϕ (F)
+

is a V–Cauchy filter. Hence, if U and V
are equivalent pre–uniform bases on X, then (X, U) and (X, V) have the same
Cauchy filters.

Definition 3.5. For every Cauchy filter F in a pre–uniform space (X, U),
define:

F ′ = {ST (F, α) : F ∈ F, α ∈ U}
+

F r =
{
S∗T (F, α) : α ∈ U

}+

F rr =
{
S∗∗T (F, α) : α ∈ U

}+
,

where

S∗T (F, α) = ∪ {A ∈ α : A ∩ F 6= ∅ for every F ∈ F}



218 A. Garćıa–Máynez and R. Mancio

and
S∗∗T (F, α) = ∪ {A : A ∈ F ∩ α} .

Remark 3.6. If F is a Cauchy filter in (X, F), then F ′, F r, F rr are also
filters in X and we have:

F ′ ⊆ F r ⊆ F rr ⊆ F.

Definition 3.7. Let F be a Cauchy filter in a pre–uniform space (X, U).

1) F es minimal if F does not properly contain any other Cauchy filter in
(X, U).

2) F is weakly round if F = F rr.

3) F is round if F = F r.
4) F is strongly round if F = F ′.

We summarize in a Theorem the most important relations among these
different kinds of filters. The proofs can be found in [2].

Theorem 3.8.

1) Every strongly round filter is round.
2) Every round filter is weakly round.
3) Every weakly round filter is minimal.
4) Two round filters F1, F2 in a pre–uniform space mingle iff F1 = F2.
5) Every neighborhood filter is weakly round.
6) If (X, U) is a semi–uniform space and if F is a Cauchy filter in (X, U),

then F ′ is also a Cauchy filter in (X, U) and F ′ is contained in any
Cauchy filter G contained in F. In fact, F ′ is a strongly round filter
in (X, U) and F ′′ = F ′.

Definition 3.9. A pre–uniform space (X, U) is complete if every Cauchy filter

in (X, U) converges.

Lemma 3.10. Let ϕ : (X, U) → (Y, V) be a unimorphism between pre–uniform
spaces and let η be a weakly round filter in X. Then ϕ (η) = {ϕ (N) : N ∈ η} is
a weakly round filter in (Y, V).

Proof. Fix an element N ∈ η and let α ∈ U be such that:

H (α) = ∪ {L : L ∈ η ∩ α} ⊆ N.

Let β ∈ V be such that β ≤ ϕ (α) = {ϕ (A) : A ∈ α}. We shall prove that
K (β) = ∪ {B : B ∈ ϕ (η) ∩ β} ⊆ ϕ (N). For each B ∈ β select an element AB ∈
α such that B ⊆ ϕ (AB). Therefore, if also B ∈ ϕ (η), we have ϕ (AB) ∈ ϕ (η)
and AB ∈ η (recall ϕ is a bijective map). Therefore,

K (β) ⊆ ∪ {ϕ (A) : A ∈ η ∩ α} = ϕ (H (α)) ⊆ ϕ (N) .

�



Completions of pre–uniform spaces 219

Lemma 3.11. Let η be a weakly round filter in a pre–uniform space (X, U)
and let A ⊆ X. Then η|A is a weakly round filter in (A, UA).

Proof. Obvious. �

Definition 3.12. A complete pre–uniform space (Y, V) is a completion of a
pre–uniform space (X, U) if there exists a unimorphic embedding ϕ : (X, U) →
(Y, V).

Definition 3.13. A pre–uniform space (X, U) is proper if every Cauchy filter

in (X, U) contains a weakly round filter.

Note 3. Let (X, U1) be a T0–pre–uniform space and choose an open pre–

uniformity basis U equivalent to U1. Let X̂ = {ξ : ξ is a weakly round filter in (X, U)}.
For every A ⊆ X and every α ∈ U, define:

 =
{

ξ ∈ X̂ : A ∈ ξ
}

α̂ =
{

 : A ∈ α
}

and let Û = {α̂ : α ∈ U}. For every x ∈ X, let ϕ (x) = ηx = filter of TU –
neighborhoods of x.

Theorem 3.14. Keep the notation of 3 and suppose the topology TU is T0.

Then Û is an open pre–uniformity basis in X̂ and ϕ is a unimorphic embed-

ding of (X, U) into
(
X̂, Û

)
. Besides, every weakly round filter in

(
X̂, Û

)
is

convergent.

Proof. Everything is proved in [2], except the last part. Let F be a weakly

round filter in
(
X̂, Û

)
. Define:

η =
{
A ∈ ∪ {α : α ∈ U} : Â ∈ F

}+
.

It is easy to prove that η is a Cauchy filter in (X, U). We shall prove that
η is weakly round. Choose an element L ∈ η. By the definition of η, there

exists a cover α ∈ U and an element A ∈ α such that A ⊆ L and  ∈ F.

Since F is weakly round in
(
X̂, Û

)
, there exists a cover β ∈ U such that

∪
{

B̂ : B̂ ∈ F ∩ β̂
}
⊆ Â. Take an element B ∈ η ∩ β. Therefore, B̂ ∈ F ∩ β̂ and

B̂ ⊆ Â. Conversely, B = ϕ−1
(
B̂

)
⊆ ϕ−1

(
Â

)
= A ⊆ L. Therefore, η is weakly

round in (X, U) and η ∈ X̂. It remains to prove that F → η. Take a set W ∈ T
Û

containing η. Since ∪ {α̂ : α ∈ U} is a basis for the topology T
Û
, we may assume

that W coincides with Â, where A ∈ α for some α ∈ U. This implies that A ∈ η

and, therefore, Â ∈ F. Hence, the filter of T
Û
–neighborhoods of η is contained

in F and F → η. �



220 A. Garćıa–Máynez and R. Mancio

Theorem 3.15. Let (X, U) be a T0–pre–uniform space. Then (X, U) admits
at least one completion iff (X, U) is proper.

Proof.

(⇒) Let (Y, V) be a completion of (X, U) and let ϕ : (X, U) → (Y, V) be a
unimorphic embedding. Let η be a Cauchy filter in (X, U). Then F =

ϕ (η)
+

is a Cauchy filter in (Y, V) and hence, there exists an element
y ∈ Y such that F → y. Let Fy be the filter of TV –neighborhoods of
y. By 3.8.5, Fy is a weakly round filter in (Y, V). Besides, Fy ⊆ F
because F → y. Using lemmas 3.10 and 3.11, we deduce that η0 =

ϕ−1
(
Fy| ϕ (X)

)
is a weakly round filter in (X, U) contained in η.

(⇐) We shall prove that
(
X̂, Û

)
is a completion of (X, U). Let F be a

Cauchy filter in
(
X̂, Û

)
and let:

η =

{
A ∈

⋃

α∈U

α : Â ∈ F

}+
.

It is easy to prove that η is a Cauchy filter in (X, U). Since (X, U) is
proper, there exists a weakly round filter η0 in (X, U) contained in η.

Let us prove that F → η0. Let α ∈ U and A ∈ α be such that η0 ∈ Â.

Therefore, A ∈ η0 ⊆ η and  ∈ F. Hence, every TÛ –neighborhood of
η0 belongs to F, i.e., F → η0.

�

Corollary 3.16. Let (X, U) be a T0–pre–uniform space where every Cauchy

filter in (X, U) contains a round filter. Then
(
X̂, Û

)
is a Hausdorff completion

of (X, U) and the only round filters in
(
X̂, Û

)
are the neighborhood filters.

Proof. See [2]. �

Example 3.17. Let (X, U) be a Hausdorff topological space and let U be the
family of open covers α of X such that some finite subfamily λ ⊆ α covers a
dense subset of X. Then U is an open pre–uniformity basis on X satisfying the

condition of 3.16. In this case, T = TU and the Hausdorff completion
(
X̂, Û

)

is a Hausdorff closed extension of (X, U).

Proof. See [2] and [5]. �



Completions of pre–uniform spaces 221

References

[1] A. Garćıa-Máynez and A. Támariz Mascarúa , Topoloǵıa General, México., Porrúa,
1988.

[2] R. Mancio-Toledo, Los espacios pre–uniformes y sus completaciones, Ph. D. Thesis,
Universidad Nacional Autónoma de México, 2006.

[3] K. Morita, On the simple extension of a space with respect to a uniformity, I, II, III
and IV, Proc. Japan Acad. 27 (1951), 65–72, 130–137, 166–171 and 632–636. 14(1953),
68–69, 571.

[4] K. Morita and J. Nagata, Topics in General Topology, Amsterdam, Elsevier Science
Publishers B. V., 1989.

[5] R. J. Porter and R. G. Woods, Extensions and Absolutes of Hausdorff Spaces, New
York., Springer-Verlag New York Inc., 1988.

Received March 2006

Accepted January 2007

Adalberto Garćıa–Máynez Cervantes (agmaynez@matem.unam.mx)
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area
de la Investigación Cient́ıfica, Circuito Exterior, Ciudad Universitaria, Distrito
Federal, C.P. 04510, México

Rubén S. Mancio Toledo (rmancio@esfm.ipn.mx)
Escuela superior de F́ısica y Matemáticas del Instituto Politécnico Nacional,
Edificio No. 9, Unidad Profesional Adolfo López Mateos, Colonia Lindavista,
México D.F., C.P. 07738, México