

@
Applied General Topology

c© Universidad Politécnica de Valencia
Volume 13, no. 1, 2012

pp. 21-25

Extending maps between pre-uniform spaces

Adalberto Garćıa-Máynez and Rubén Mancio-Toledo

Abstract

We give sufficient conditions on a uniformly continuous map
f : (X, U) → (Y, V ) between completable T1-pre-uniform spaces (X, U),
(Y, V ) to have a continuous or a uniformly continuous extension
̂f : ̂X → ̂Y between the corresponding completions.

2010 MSC: 54A20, 54E15

Keywords: Minimal, round, pre-uniform, completion, extension

1. Preliminary results

The basic concepts used in this paper: pre-uniformity bases, Cauchy or
minimal filters, round, weakly round or strongly round filters and completion
conditions are given in [1]. The concept of pre-uniform basis appeared in 1970
under the name of structure [3]. However, non Hausdorff pre-uniform spaces
were very seldom considered in Harris monography.
T1-pre-uniform spaces have an important property: Every Cauchy filter con-

tains a unique weakly round filter and every neighborhood filter is weakly

round. The set of weakly round filters X̂ of a T1-pre-uniform space has a com-

plete T1-pre-uniform basis Û such that the map h: (X,U) → (X̂,Û) which
assigns to each x ∈ X its neighborhood filter is a uniform embedding. Hence,
any uniformly continuous map ϕ: (X,U) → (Y,V ) between T1-pre-uniform
spaces induces a map ϕ̂: (X̂,Û) → (Ŷ , V̂ ) which sends every weakly round
filter ξ ∈ X̂ into the unique weakly round filter N in Ŷ which is contained in
the Cauchy filter

ϕ(ξ) = {ϕ(L) | L ∈ ξ}+ .
(For every subfamily G of the power set of a set Z, we define G+ = {L ⊆ Z |
for some G ∈ G,G ⊆ L}). If k : (Y,V ) → (Ŷ , V̂ ) is the canonical uniform


22 A. Garćıa-Máynez and R. Mancio-Toledo

embedding, i.e. k(y) = neighborhood filter of y, we have the relation ϕ̂ ◦ h =
k ◦ ϕ. In this paper, we find conditions on ϕ, U and V which insure that ϕ̂ is
continuous or uniformly continuous.

2. Main results

We start this section with a lemma.

Lemma 2.1. Let (X,U) be a T1-pre-uniform space and suppose (X,τU ) is a
T1-space. Then every Cauchy filter ξ in (X,U) contains a unique minimal filter
ξ′.

Proof. We know ξ′ = {ST ∗∗(ξ,α) | α ∈ U}+ is U-minimal and is contained in
ξ, where

ST
∗∗(ξ,α) =

⋃
{L | L ∈ α ∩ ξ} .

Suppose N ⊆ ξ is another U-minimal filter.
Therefore, N ′ = N ⊆ ξ′. The minimal property of ξ′ implies that N ′ =

N = ξ′. �

We give two cases in which ϕ̂ is uniformly continuous.

Lemma 2.2. Suppose for each ξ ∈ X̂ − h(X),ϕ(ξ) = ϕ(ξ)′. Then ϕ̂ is uni-
formly continuous.

Proof. Let β ∈ V and let α ∈ U be such that α ≤ ϕ−1(β). We shall prove
that α̂ ≤ ϕ̂−1(β̂ ). Let A ∈ α and B ∈ β be such that A ⊆ ϕ−1(B). We claim
that Â ⊆ ϕ̂−1(B̂). Let us take ξ ∈ Â. Then A ∈ ξ and ϕ(A) ∈ ϕ(ξ). Since
ϕ(A) ⊆ B, we have also B ∈ ϕ(ξ). Therefore, ϕ̂(ξ) = ϕ(ξ) ∈ B̂ and the proof
is complete. �

Lemma 2.3. If (Y,V ) is a semi-uniform space, ϕ̂ is uniformly continuous.

Proof. Let β ∈ V . Since (Y,V ) is a semi-uniform space, there exists a cover
γ ∈ V which satisfies the following condition:
Su) For each C ∈ γ, there exists δC ∈ V and BC ∈ β such that ST (C,δC)
⊆ BC.

Let α ∈ U be such that α ≤ ϕ−1(γ). We shall prove that α̂ ≤ ϕ̂ −1(̂β). If
A ∈ α, there exists a set C ∈ γ such that A ⊆ ϕ−1(C). By condition Su),
there exist δC ∈ V and BC ∈ β such that ST (C,δC) ⊆ BC. We claim that
Â ⊆ ϕ̂−1(B̂C). If ξ ∈ Â, we have A ∈ ξ. Since ϕ(A) ⊆ C, we have C ∈ ϕ(ξ)+.
Therefore, ST (C,δC) ∈ ϕ(ξ)′ = ϕ̂(ξ). Since ST (C,δC) ⊆ BC, we conclude that
BC ∈ ϕ̂(ξ) and ϕ̂(ξ) ∈ B̂C. �

Lemma 2.4. Let X,Y be T2-spaces and let U,V , respectively, be the families of
densely finite covers of X,Y. Let ϕ: X → Y be continuous, open and surjective.
Then ϕ is uniformly continuous as a map from (X,U) onto (Y,V ).


Extending maps between pre-uniform spaces 23

Proof. Let β be a densely finite cover of Y . Then we can find a finite subfamily
{B1,B2, · · · ,Bn} ⊆ β such that B−1 ∪ B

−
2 ∪ · · · ∪ Bn− = Y. If B = B1 ∪ B2 ∪

· · · ∪ Bn and α = f−1(β), the hypotheses imply that
{A1,A2, . . . ,An} ⊆ α

where Ai = f
−1(Bi) for i = 1,2, . . . ,n, satisfies

A−1 ∪ A
−
2 ∪ · · · ∪ A

−
n = f

−1(B−) = X .

Hence α is a densely finite cover of X and α ≤ f−1(β) (In fact, α = f−1(β)).
Then f is uniformly continuous. �

Lemma 2.5. Keep the hypotheses of (2.4). Then ϕ̂: X̂ → Ŷ is continuous and
surjective.

Proof. Let N ∈ X̂ and let T be an open set in Y such that ϕ̂(N) = ϕ(N)′ ∈
T̂ . Then T ∈ ϕ(N)′. Therefore, there exists a cover γ ∈ V such that T ⊇
ST

∗∗(ϕ(N),γ). Since ϕ: (X,U) → (Y,V ) is uniformly continuous (2.4), the
filter ϕ(N) is Cauchy in (Y,V ). Select an element N0 ∈ γ ∩ ϕ(N). Then
ϕ−1(N0) ∈ ϕ−1(γ) ∩ N and N0 ⊆ ST ∗∗(ϕ(N),γ) ⊆ T . Therefore:

ϕ−1(N0) ⊆ ST ∗∗(N ,ϕ−1(γ)) = ϕ−1(ST ∗∗(ϕ(N),γ)) ⊆ ϕ−1(T).
We shall prove that ϕ̂(ST (N ,ϕ−1(γ)̂ ) ⊆ T̂ and the continuity of ϕ̂ will

follow.
Let M ∈ ST (N ,ϕ−1(γ)̂ ). Then there exists an element C ∈ γ such that

ϕ−1(C) ∈ M ∩ N . Hence, ϕ−1(C) ⊆ ST ∗∗(N ,ϕ−1(γ)) ⊆ ϕ−1(T) and C ⊆
ST

∗∗(ϕ(N),γ) ⊆ T . We also have

C ⊆ ST ∗∗(ϕ(M),γ) ∈ ϕ(M)
′
= ϕ̂(M).

Then T ∈ ϕ̂(M) and ϕ̂(M) ∈ T̂ . �
Before we prove ϕ̂ is surjective, we need a lemma.

Lemma 2.6. A non-adherent filter T in (X,U) is U-round if and only if T
has as a basis an ultrafilter of open sets.

Proof. Suppose T is a non-adherent round filter in (X,U). Let G be the family
of open sets in T and take an open set V such that V ∩G 
= ∅ for every G ∈ G.
We have to prove that V ∈ T and that will convert G into an ultrafilter of open
sets.

Since T is non-adherent, the family {X − F−|F ∈ T } is an open cover of
X. Hence.

α = {V,X − V −} ∪ {X − F−|F ∈ T }
is a densely finite cover of X. Since T is U-Cauchy, we have V ∈ T or
X − V − ∈ T . If we had X − V − ∈ T , we use the roundness of T and find
a cover β ∈ U such that X − V − ⊇ ST ∗(T ,β) = ∪{B ∈ β | B ∩ F 
= ∅ for
every F ∈ T }. If G ∈ β ∩ T , we have G ⊆ X − V − and hence V ∩ G = ∅, a
contradiction. Therefore we must have V ∈ T and G is an ultrafilter of open
sets.


24 A. Garćıa-Máynez and R. Mancio-Toledo

Conversely, suppose G is an ultrafilter of open sets. We have to prove that
T is U-round. We prove first that T is U-Cauchy. Let α ∈ U. If T ∩ α = ∅,
then A /∈ T ∩ τ for every A ∈ α. Let {A1,A2, . . . ,An} ⊆ α be such that
X = A1

− ∪A2− ∪· · ·∪An−. Since Ai /∈ T ∩ τ and T ∩ τ is an ultrafilter of open
sets, we can find elements Gi ∈ T ∩ τ such that Ai ∩ Gi = ∅ (i = 1,2, . . . ,n).
Hence (G1 ∩ G2 ∩ · · · ∩ Gn) ∩ (A1 ∪ A2 ∪ · · · ∩An) = ∅. But A1 ∪ A2 ∪ · · · ∪ An
is dense in X. Hence G1 ∩ G2 ∩ · · · ∩ Gn = ∅, a contradiction. We finally
prove that T is U-round. Pick any element F0 ∈ T and consider the cover
α = {F0} ∪ {X − F− | F ∈ T }. Clearly ST ∗(T ,α) = F0 and hence T is
U-round. �

In [4] it is proved that every Cauchy filter in (X,U), where U is the family
of densely finite covers of the Hausdorff space (X,τ), contains an U-round
filter and by [1],(X,U) has a completion (X̂, Û ) where every Û-round filter is
convergent and the topologyτ

̂U is Hausdorff closed. Besides the completion
(X̂, Û ),(X,τ) has the Katetov extension kX, which is also Hausdorff closed.
In this volume we prove that in general, the extensions X̂ and kX are non-
equivalent.

3. Applications

Proposition 3.1. Let X be a separable, metrizable, dense in itself, 0-dimensional
space and let Z be a compact, Hausdorff, separable space. Then there exists a

surjective continuous map g : X̂ → Z, where X̂ is the completion of the pre-
uniformity basis of X consisting of all densely finite covers of X.

Proof. The hypothesis imply the existence of mutually disjoint non-empty open

sets L1,L2, . . . such that X =
∞
∪

n=1
Ln. The map ϕ: X → N where Ln = ϕ−1(n)

for each n ∈ N, is continuous, open and surjective. By 2.4, there exists a
continuous surjective extension ϕ̂: X̂ → N̂. But N̂ coincides with the Stone-
Čech compactification βN of N (because a cover α of N is densely finite if and
only if it is finite). On the other hand, by the universal property of βN, there
exists a continuous surjective map ψ : βN → Z. Hence, g = ψ◦ϕ̂ is a continuous
surjective map from X̂ onto Z. �

Proposition 3.2. Let X be a non-empty completely metrizable separable space.

Then there exists a continuous surjective map ψ : (N
w

)̂→ X̂.

Proof. N
w

may be identified with the set of irrational numbers and this space
satisfies the conditions of (3.1). On the other hand, there exists a continuous
open surjective map ϕ: N

w

→ X (see 5.15 in [2]). Using 2.4, we complete the
proof. �

Corollary 3.3. If Z is a Tychonoff separable space which is either com-
pact or completely metrizable, then there exists a continuous surjective map

ψ : (Nw)̂ → Ẑ.


Extending maps between pre-uniform spaces 25

We finish this paper with a problem:

Problem 3.4. Is every Čech-complete separable space a continuous image of
(N

w

)̂ ?

References

[1] A. Garćıa-Máynez and R. Mancio Toledo, Completions of pre-uniform spaces, Appl.
Gen. Topol. 8, no. 2 (2007), 213–221.

[2] A. Garćıa-Máynez and A. Tamariz, Topoloǵıa General, Ed. Porrúa, México, 1988.
[3] D. Harris, Structures in Topology, Memoirs of The American Mathematical Society, No.

115, Providence, Rhode Island, AMS, 1971.

[4] R. Mancio Toledo, Los espacios pre-uniformes y sus completaciones, Ph. D. Thesis,
UNAM, 2006.

(Received January 2010 – Accepted January 2011)

A. Garćıa-Máynez (agmaynez@matem.unam.mx)

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Área
de la Investigación Cient́ıfica, Circuito Exterior, Ciudad Universitaria, 04510
México, D.F. México

Rubén Mancio-Toledo (rmancio@esfm.ipn.mx)
Escuela Superior de F́ısica y Matemáticas, Instituto Politécnico Nacional, Unidad
Profesional Adolfo López Mateos, Col. Lindavista, 07738 México, D.F.


	Extending maps between pre-uniform spaces. By A. García-Máynez and R. Mancio-Toledo