GMaynezAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 8, No. 2, 2007

pp. 223-237

A Survey on Wallman Bases

Adalberto Garćıa-Máynez C.

Abstract. Wallman bases are frequently used in compactifica-

tion processes of topological spaces. However, they are also related

with quasi–uniform structures and they are useful to characterize some

topological properties. We present a brief survey on the subject which

supports these statements.

2000 AMS Classification: 54C25, 54C45, 54D35, 54D80.

Keywords: Wallman basis, annular basis, ultrafilter, perfect extension, Wall-
man type, regular Wallman, equivalent compactifications, cover uniformity ba-
sis, quasi–uniformity, transitive, totally bounded, symmetric, point symmetric,
locally symmetric.

1. Historical Background

Among compactification methods in topology, we bring out two of them,
which are perhaps the most useful:

• Wallman’s method of ultrafilters.
• Completion of totally bounded uniform spaces.

The main advantadge of the method of ultrafilters is its generality: it may
be applied to any T1-space. Wallman applied this method for the first time in
1938 ([17]) and he proved that any T1–space X has a T1–compactification ωX
which coincides with the Stone–Čech compactification if X is normal.

The concept of uniform space was introduced by A. Weil, in 1937 ([18]) and
he proved, among other things, that every uniform space (X, U) has unique
uniform completion (unique up to uniform equivalences). The notion of totally
bounded uniform spaces was formally introduced by N. Bourbaki in 1940, al-
though it was used implicitly by Weil in the well known characterization of
compactness in uniform spaces:

A uniform space (X, U) is compact (i.e., the topology TU in-
duced by U is compact) iff (X, U) is complete and satisfies an
extra condition which is precisely total boundedness.



224 A. Garćıa-Máynez

Since total boundedness is not lost under uniform extensions, we conclude
that the uniform completion of a totally bounded uniform T1–space X is a
compactification of X.

The method of Wallman can be applied in smaller families of closed sets and
produce a compactification in exactly the same way. One of the requirements
is that the family of complements of the given closed sets is a basis for the
topology of the space.

2. Annular bases and quasi–uniformities

Every topological space (X, T ) has many bases, i.e., subfamilies B of T such
that every U ∈ T can be expressed as a union of some members of B. Annular
bases are not so numerous:

A basis B of T is annular if it satisfies two conditions:

i) ∅ ∈ B and X ∈ B.
ii) B, B′ ∈ B implies that B ∩ B′ ∈ B and B ∪ B′ ∈ B.

It is possible that T is the only annular basis of itself. A less strong con-
dition is that T has a minimal annular basis, i.e., an annular basis which is
contained in every other annular basis. This happens in any locally compact
0–dimensional space.

A successful generalization of uniform spaces was found in 1960 ([1]). Only
completely regular spaces admit uniform structures which produce their topol-
ogy. However, any topological space admits compatible quasi–uniform struc-
tures. We briefly state the main definitions.

A quasi–uniformity U on a set X is a filter in X × X satisfying the following
properties:

i) The diagonal △ (X) = {(x, x) | x ∈ X} is contained in each member of
U, i.e., each U ∈ U is a reflexive relation in X, and

ii) whenever U ∈ U, there exists an element V ∈ U such that V ◦ V ⊆ U.

A connector on X is, by definition, a reflexive relation on X. Therefore, every
element of a quasi–uniformity is a connector and the family of all connectors
on X is the largest quasi–uniformity on X.

Each quasi–uniformity U on a set X determines a topology TU on X according
to the following definition:

∗) A subset A of X belongs to TU iff for every x ∈ A, we may find an
element Ux ∈ U such that:

Ux (x) = {y ∈ X | (x, y) ∈ Ux} ⊆ A.

A quasi–uniformity U on a topological space (X, T ) is compatible (with T )
if T = TU .

A connector U on a topological space (X, T ) is a neighbornet if for each
x ∈ X, U (x) = {y ∈ X | (x, y) ∈ U} is a T –neighborhood of x. A sequence
U1, U2, . . . of neighbornets of a topological space (X, T ) is normal if for every
n ∈ N, we have U2n+1 = Un+1 ◦ Un+1 ⊆ Un. A neighbornet U is normal if



Wallman Bases 225

it belongs to a normal sequence of neighbornets. Clearly, if U is a compatible
quasi–uniformity on a topological space (X, T ), each member U ∈ U is a normal
neighbornet of (X, T ).

In an arbitrary topological space (X, T ) there exists at least one compatible
quasi–uniformity, the Pervin quasi–uniformity (see [9]). Among all compatible
quasi–uniformities on a topological space (X, T ), there exists one, the fine
quasi–uniformity, which contains every other compatible quasi–uniformity:
take all normal neighbornets on (X, T ). Spaces where the fine quasi–uniformity
is the only compatible quasi-uniformity are characterized in [8]. Such spaces
have only one annular basis, the topology itself.

A basis η for a quasi–uniformity U is simply a filterbase in X × X which
generates U. A quasi–uniformity basis η is transitive if each element N ∈ η is
transitive (i.e., N ◦ N ⊆ N) and symmetric if each N ∈ η is symmetric (i.e.,

N = N−1). A connector A on a set X is totally bounded if there exists a finite
cover {E1, E2, . . . , Es} of X such that Ei × Ei ⊆ A for every i = 1, 2, . . . , s. A
quasi–uniformity U on X is totally bounded if each member of U is totally
bounded. A quasi–uniformity U on X is transitive (resp., a uniformity) if U
has a transitive (resp., a symmetric) basis. The Pervin quasi–uniformity on a
topological space is both transitive and totally bounded.

An important fact observed by Romaguera and Sánchez Granero is the fol-
lowing:

There exists a bijection between the family of all transitive and totally
bounded compatible quasi–uniformities on a topological space (X, T ) and the
family of all annular bases of T . This bijection, which respects inclusions, is
described in [10]. Of course, the quasi-uniformity corresponding to T is the
Pervin quasi–uniformity.

3. Wallman bases

A Wallman basis B on (X, T ) is an annular basis of T satisfying the extra
condition:

WB) If x ∈ B ∈ B, there exists a closed set H such that x ∈ H ⊆ B and such
that X − H ∈ B.

Whenever we restrict ourselves to a definite basis B of a topological space
(X, T ), by a cobasic set we simply mean a closed set H ⊆ X such that
X − H ∈ B. C (B) denotes the family of all cobasic sets with respect to B. So,
condition WB) is equivalent to say that every basic set B ∈ B is a union of
cobasic sets H ∈ C (B).

A quasi–uniformity U on a set X is said to be point symmetric if for each
U ∈ U and each x ∈ X, there exists a symmetric connector Vx ∈ U such that
Vx (x) ⊆ U (x) (i.e., (x, y) ∈ Vx implies (x, y) ∈ U).

In the correspondence between transitive totally bounded compatible quasi–
uniformities of a topological space (X, T ) and the family of annular bases of
T , if we restrict ourselves to point–symmetric quasi–uniformities, we have a
bijection with the family of all Wallman bases of (X, T ) (see [10]).



226 A. Garćıa-Máynez

Not every topological space (X, T ) admits Wallman bases. However, a very
mild restriction on T guarantees their existence:

Theorem 3.1. A topological space (X, T ) admits Wallman bases iff T itself is
a Wallman basis, i.e., iff every open set U ∈ T is a union of closed sets.

The condition in the Theorem above is equivalent to the following property:

R0) If x, y ∈ X and y ∈ Cℓ ({x}), then Cℓ ({y}) = Cℓ ({x}).

Spaces satisfying condition R0) are simply called R0–spaces. It is obvious
that every T1–space (i.e., spaces where every finite set is closed) is R0. In fact,
the condition T1 is equivalent to the conditions R0 and:

T0) If x, y ∈ X and y ∈ Cℓ ({x}) and x ∈ Cℓ ({y}), then x = y.

A Wallman basis B on (X, T ) is regular if whenever x ∈ B ∈ B, there exist

elements D ∈ B and H ∈ C (B) such that x ∈ D ⊆ H ⊆ B.
A quasi–uniformity U on X is locally symmetric if for each U ∈ U and

x ∈ X, there exists a symmetric element Vx ∈ U such that V
2
x (x) ⊆ U (x).

Going back to the correspondence Theorem, we obtain:

Theorem 3.2. The quasi–uniformity UB which corresponds to a Wallman basis
B of (X, T ) is locally symmetric iff B is regular.

Normality is perhaps the most widely used property in General Topology; it

may understood in at least four different ways: a) Topological spaces, b) open
covers, c) connectors and d) Wallman bases.

Of course, the first is the most important: a topological space is normal
if for every pair H K of disjoint closed sets, there exist disjoint open sets UH,
UK such that H ⊆ UH and K ⊆ UK. The last one is closely related to this: a
Wallman basis B of a topological space (X, T ) is normal if for every pair H,
K of disjoint cobasic sets, there exist disjoint basic sets BH, BK ∈ B such that
H ⊆ BH and K ⊆ BK. Combining these two definitions, we have:

An R0–space (X, T ) is normal iff the Wallman basis T is nor-
mal.

However a space with a normal Wallman basis may not be normal. In fact,
every locally compact Hausdorff space (X, T ) has a normal Wallman basis:
define B as the family of open sets B such that Cℓ (B) or X − B is compact.
More generally, if (X, T ) is a Hausdorff space and the family B of open sets
V ∈ T with compact boundary is a basis of T , then B is a normal Wallman
basis of (X, T ). These spaces were called peripherally compact by Gordon
T. Whyburn and it is quite clear that every locally compact Hausdorff space
is peripherally compact but the converse may not be true. There are many
examples in the literature of locally compact Hausdorff spaces which are not
normal (see [12], Ex. 106 and [2], Ex. 2, p.239). On other side, every completely
regular space (X, T ) has a normal Wallman basis, the collection of cozero sets:
V ⊆ X is a cozero set if there exists a continuous function:

f : (X, T ) → (R, Td) (Td usual topology of R)



Wallman Bases 227

such that V = f −1 (R − {0}) = {x ∈ X | f (x) 6= 0}.
A remarkable property of normal Wallman basis is the existence of scales

between any pair of disjoint cobasic sets:

Definition 3.3. Let A, B be disjoint subsets of a topological space (X, T ) and

let D =

{
m

2n
| m, n ∈ N

}
be the family of dyadic rationals in the open unit

interval (0, 1). A scale between A and B is a map ϕ : D → P (X) such that
whenever d1, d2 ∈ D, d1 < d2, we have CℓX (ϕ (d1)) ⊆ IntX (ϕ (d2)).

Every scale ϕ : D → P (X) determines a continuous map f : X → [0, 1 ] from
X to the closed unit interval such that:

f −1 (0) = ∩ {ϕ (r) | r ∈ D}

and
f −1 (1) = X − ∪ {ϕ (r) | r ∈ D} .

The map f is defined exactly as in the proof of Urysohn’s Lemma:

f (x) =

{
0 if x ∈ ∩ {ϕ (r) | r ∈ D}

sup {r ∈ D | x /∈ ϕ (r)} otherwise
.

If B is a normal Wallman basis of a topological space and if H, K are disjoint
cobasic sets, apply the normality of B and obtain two disjoint basic sets BH,
BK such that H ⊆ BH and K ⊆ BK. Proceed to define the scale ϕ by induction

on the power of two in the denominator of a dyadic rational
m

2k
∈ D, starting

with ϕ
(

1

2

)
= BH. Inductively, if ϕ has already been defined

in:

Dn =

{
m

2k
∈ D | k ≤ n

}

and in such a way that for each d ∈ Dn we have ϕ (d) ∈ B and H ⊆ ϕ (d) ⊆ Fd
for some cobasic set Fd contained in X−K and such that whenever d1, d2 ∈ Dn,
d1 < d2, there exists a cobasic set F such that:

(3.1) ϕ (d1) ⊆ F ⊆ ϕ (d2) ,

we proceed to define ϕ on Dn+1: let
m

2n+1
∈ Dn+1, where m is odd and

1 < m < 2n+1 − 1. Then d1 =
m − 1

2n+1
and d2 =

m + 1

2n+1
both belong to

Dn. By the induction hypothesis, there exists a cobasic set F such that (3.1)
holds. Then F and X − ϕ (d2) are disjoint cobasic sets. Since B is normal,
there exist disjoint basic sets B, B′ such that F ⊆ B and X − ϕ (d2) ⊆ B

′.

Define ϕ

(
m

2n+1

)
= B. If m = 1, we find disjoint basic sets B, B′ containing

H and X − ϕ

(
1

2n

)
and define ϕ

(
1

2n

)
= B. If m = 2n+1 − 1, we find disjoint



228 A. Garćıa-Máynez

basic sets B, B′ containing F 2n−1
2n

and K and define ϕ

(
2n+1 − 1

2n+1

)
= B. This

completes the inductive construction.
A very important consequence of this result, is that every topological space

which admits a normal Wallman basis is completely regular. Hence we have
the following characterization:

Theorem 3.4. A topological space (X, T ) is completely regular iff T admits a
normal Wallman basis. In fact, every completely regular infinite T2–space of
weight α has a normal Wallman basis of cardinality α.

Completely regular Hausdorff spaces, and only them, admit Hausdorff com-
pactifications (see [3], 3.30.3).

Every Wallman basis of a T1–space (X, T ) yields a T1–compactification of
(X, T ). To construct such a compactification, we define the concept of Wallman
ultrafilter:

Sea B be a Wallman basis of X and let C (B) be the family of cobasic sets.
A non-empty subfamily ξ of C (B) is a Wallman ultrafilter if ξ satisfies the
following conditions:

1) Each element H ∈ ξ is non-empty.
2) For every pair of elements H, K ∈ ξ, H ∩ K also belongs to ξ.
3) If H ∈ ξ and H ⊆ K ∈ C (B), then K ∈ ξ.
4) An element K ∈ C (B) belongs to ξ iff K ∩ H 6= ∅ for every H ∈ ξ.

Observe every point p ∈ X determines a Wallman ultrafilter, namely ξp =
{H ∈ C (B) | p ∈ H}.

The collection of Wallman ultrafilters (respect to B) is denoted as X (B).
There is a natural map v : X → X (B) which assigns to every p ∈ X its fixed
ultrafilter ξp. For every A ⊆ X we define a subset A

∗ of X (B) by means of the
formula:

A∗ = {ξ ∈ X (B) | for some F ∈ ξ, F ⊆ A} .

This operator A 7→ A∗ respects inclusions and for every pair of subsets C, D
of X, we have:

(C ∩ D)
∗

= C∗ ∩ D∗.

The formula (C ∪ D)
∗

= C∗ ∪ D∗ is also valid provided that C and D both
belong to B ∪ C (B) (see [3]). The family

B∗ =
{
B∗ | B ∈ B

}

is an annular basis for a compact T1–topology T
∗ of X (B). The natural map

v : (X, T ) →
(
X (B) , T ∗

)

is then injective, continuous, open onto its range v (X) and with v (X) dense in
X (B). Therefore, the pair (v, X (B)) is a T1–compactification of X, called the
Wallman compactification of X with respect to the basis B.



Wallman Bases 229

4. Topological and embedding properties

A general problem we may set ourselves is the following:

Given a topological property P and an embedding property J,
find conditions on a Wallman basis B of (X, T ) which insure
that

(
X (B) , T ∗

)
has property P or that X is J–embedded in

X (B).

We give first some definitions:

Definition 4.1. Two Wallman bases of (X, T ) are equivalent if every pair
of disjoint cobasic sets with respect to any one of the bases, are contained in
disjoint cobasic sets with respect to the other.

It is not difficult to prove that if B is a normal Wallman basis of a Tychonoff
space X and if there exists a pair of non–compact disjoint cobasic sets, then
there exists a normal Wallman basis B ′ for the same topology which is not
equivalent to B.

Definition 4.2. An extension Z of a space X is perfect if whenever we have
a separation in X:

X − K = U ∪ V, K closed in X, U, V, open and disjoint

we also have a separation in Z:

X − CℓZ (K) = U1 ∪ V1, U1 ⊇ U, V1 ⊇ V, U1, V1, open and disjoint in Z.

A simple characterization of perfect extensions can be given if we use the
operator e : TX → TZ between the topologies of X and Z, where

e (U) = Z − CℓZ (X − U) .

Then Z is a perfect extension of X iff for every pair U, V of disjoint open sets
in X, we have e (U ∪ V) = e (U) ∪ e (V).

Definition 4.3. If Z is an extension of X, we say X is locally connected in Z
if Z has a basis B such that for every B ∈ B, B ∩ X is a connected subset of X.

It is easy to see that if X is locally connected in Z, then Z is a perfect
extension of X.

It is well known that if X is a Tychonoff space, then the Stone–Čech com-
pactification β X is a perfect extension of X and X is C∗–embedded in β X.

Another important example of a perfect compactification is the Wallman
compactification of a peripherally compact Hausdorff space X, where B is the
family of open subsets of X with compact boundary.

We state a few classical results in this context:

Theorem 4.4. Let B a Wallman basis of a T1–space (X, T ). Then
(
X (B) , T ∗

)

is a Hausdorff space iff B is normal.



230 A. Garćıa-Máynez

Theorem 4.5. If B is a normal Wallman basis of a Tychonoff space (X, T ),
then X is C∗–embedded in X (B) iff B is equivalent to the cozero Wallman basis
of (X, T ).

The following results were proved by myself: ([5])

Theorem 4.6. If B is a Wallman basis of a T1–space (X, T ), then X (B) is a
perfect compactification of X iff B satisfies the following condition:

∗) If K ⊆ B, where K ∈ C (B) and B ∈ B and if L is an open set in X
such that B ∩ F r (L) = ∅, then there exists a basic set BL ∈ B such
that K ∩ L ⊆ BL ⊆ B ∩ L.

Condition ∗) is clearly satisfied if every clopen subset of a basic set is also a
basic set.

Theorem 4.7. If B is a Wallman basis of a T1–space (X, T ) satisfying the
property:

(4.1) B ∈ B, E component of B ⇒ E ∈ B,

then X is locally connected in X (B) iff B satisfies the following condition:

∗∗) If K ⊆ B, where K ∈ C (B) and B ∈ B, then there exists a finite
collection C1, C2, . . . , Cn of connected subsets of X such that:

K ⊆

n⋃

i=1

Ci ⊆ B.

We say B is locally connected if it satisfies the property 4.1.
Condition ∗∗) is satisfied if X is locally connected, Hausdorff and peripherally

compact and X has only a finite number of components.
Before stating two more results, we need a definition:

Definition 4.8. A Hausdorff compactification Z of a space X is of
Wallman type if X has a Wallman basis B such that Z and X (B) are equivalent
compactifications of X.

We have then:

Theorem 4.9 ([14]). If Z is a compact metrizable space and if X is dense in
Z, then Z is a Wallman type compactification of X.

Theorem 4.10 ([5]). If Z is a compact and Hausdorff and if X is Gδ–dense
in Z (i.e., every non-empty Gδ set in Z intersects X) then Z is a Wallman type
compactification of X.

Before going on, we need some more definitions:

Definition 4.11. A topological space (X, T ) is S–metrizable if there exists a
metric d on X inducing T and satisfying the following property:

S) For every ε > 0, there exists a finite cover C1, C2, . . . , Cn of X consist-
ing of connected sets of diameter < ε.



Wallman Bases 231

Every S–metrizable space is locally connected, separable and has only a finite
number of connected components.

We characterize now S–metrizability:

Theorem 4.12. The following three properties of a topological space (X, T )
are equivalent:

a) X is S–metrizable.
b) X has a perfect locally connected metrizable compactification Z ([4]).
c) T has a countable normal locally connected Wallman basis consisting

of open domains. (This equivalence is easily obtainable from results
of [14]).

Another topological concept which leads to many open problems is weak
pseudocompactness:

Definition 4.13. A Tychonoff space (X, T ) is weakly pseudocompact if (X, T )
has a Hausdorff compactification in which X is Gδ–dense.

A non-compact locally compact T2–Lindelöf space X cannot be weakly pseu-
docompact. Obviously, every Tychonoff pseudocompact space is weakly pseu-
docompact. The Hedgehog J (α), where α ≥ ℵ1 is an example of a weakly
pseudocompact space which is not pseudocompact. As far as I know, it is an
open problem if Rα (α ≥ ℵ1) is weakly pseudocompact.

We have however, the following characterization:

Theorem 4.14 ([6]). A Tychonoff space (X, T ) is weakly pseudocompact iff
T has a normal Wallman basis B such that every countable cover of X with
elements of B has finite subcover.

5. Cover uniformities and Wallman bases

There is a strong relation between normal Wallman bases and cover
uniformities. The treatment of uniform spaces thru covers instead of connectors
must be seen as an alternative, but the two treatments are equivalent.

Definition 5.1. A cover uniformity basis on a set X is a non–empty family
G of covers of X such that for every pair of covers α, β ∈ G, there exists a
cover γ ∈ G which refines baricentrically each of the covers α, β, i.e., for
every p ∈ X we may find elements Ap ∈ α, Bp ∈ β such that ST ( p, γ) =
∪ {C | p ∈ C ∈ γ} ⊆ Ap ∩ Bp. A cover uniformity on X is simply a cover
uniformity basis G on X with the additional property:

∗) If α ∈ G and if β is a cover of X refined by α, then β ∈ G.

Each cover uniformity basis G on X is contained in a unique smallest cover
uniformity on X, namely G ⊆ G+ where

G+ = {β | β is a cover of X and some cover α ∈ G refines β} .



232 A. Garćıa-Máynez

Two cover uniformity bases G1, G2 on X are equivalent if G
+
1 = G

+
2 . Each

cover uniformity basis G on X determines a topology TG on X defined by:

L ∈ TG ⇔ ∀x ∈ L, ∃ αx ∈ G ⋔ ST (x, αx) ⊆ L.

It is easy to see that two equivalent cover uniformity bases determine the
same topology on X but the converse is not true in general.

A standard result (but not so obvious), states that for every cover uniformity
basis G on a set X, TG is a completely regular topology on X. Besides, TG is
T1 (and hence, TG is a Tychonoff topology on X) iff for every x ∈ X, we have
∩ {ST (x, α) | α ∈ G} = {x}.

In our terminology, a cover uniform space is a pair (X, G), where G is a cover
uniformity basis on X. It is easy to see that for every subset A ⊆ X, G|A ={
α|A | α ∈ G

}
is a cover uniformity basis on A and

(
A, TG|A

)
is a subspace of

(X, TG). So, by a cover uniform subspace of (X, G), we simply mean a pair(
A, G|A

)
, where A ⊆ X.

The correspondence between the concepts of cover uniform space and uni-
form space is very simple:

Given a cover uniform space (X, B) we construct a filter basis FB of symmet-
ric connectors of X, namely, each α ∈ B determines the symmetric connector:

E (α) = ∪ {L × L | L ∈ α} .

The filter F +B = {T ⊆ X × X | E (α) ⊆ T for some α ∈ B} is a uniformity on
X and the topologies TB, TFB coincide. Conversely, given a symmetric quasi–
uniformity basis F on X, each F ∈ F determines an indexed cover:

αF = {F (x) | x ∈ X} .

The family
B = {αF | F ∈ F}

is a cover uniformity basis: let αF1 , αF2 ∈ B and let G ∈ F be such that
G2 ⊆ F1 ∩ F2. Then

α
△
G = {ST (x, αG) | x ∈ X}

refines both covers αF1 , αF2 : in fact, if z ∈ ST (x, αG) and G (y) contains both
points x, z, then (y, z) , (y, x) ∈ G and so (x, y) ∈ G−1 = G and (x, z) ∈ G2 ⊆
F1 ∩ F2 and z ∈ F1 (x) ∩ F2 (x). In this case we have also the same topologies
TB = TF . �

A map ϕ : (X, G) → (Y, H) between cover uniform spaces is uniformly
continuous if for every ε ∈ H we may find a cover δ ∈ G such that δ refines
ϕ−1 (ε) =

{
ϕ−1 (E) | E ∈ ε

}
. ϕ is a unimorphism if ϕ es bijective and both

maps ϕ : (X, G) → (Y, H), ϕ−1 : (Y, H) → (X, G) are uniformly continuous.
ϕ : (X, G) → (Y, H) is a unimorphic embedding if ϕ is a unimorphism from

ϕ : (X, G) →
(
ϕ (X) , H|ϕ (X)

)
and if ϕ (X) is dense in Y. It is obvious that

if G1, G2 are cover uniformity bases on X, then the identity map j : (X, G1) →
(X, G2) is a unimorphism iff G1 and G2 are equivalent.

We have several important concepts in uniform space theory:



Wallman Bases 233

Definition 5.2.

a) A filter F on a cover uniform space (X, G) is Cauchy if F ∩ α 6= ∅
for every α ∈ G.

b) A cover uniform space (X, G) is complete if every Cauchy filter on
(X, G) is convergent (with respect to the topology TG).

c) A cover uniform space (X, G) is totally bounded if every cover α ∈ G
has a finite subcover for X.

d) A cover uniform space (Y, H) is a cover completion of a cover uniform
space (X, U) if (Y, H) is complete and if there exists a unimorphic
embedding ϕ : (X, U) → (Y, H).

The easiest example of a completion is the metric completion: Let (X, d) be

a metric space and let
(
X̃, d̃

)
be a metric completion of (X, d). For each ε > 0

define

αε =
{
V dε (x) | x ∈ X

}
;

α̃ε =
{
V d̃ε (x) | x ∈ X

}
.

Then Gd = {αε | ε > 0} is a cover uniformity basis on X, Gd̃ = {α̃ε | ε > 0}

is a cover uniformity basis on X̃ and
(
X, G

d̃

)
is a cover completion of (X, Gd ).

We state without proof a few important theorems on cover uniform space
theory. (See [3], Chap. 7).

Theorem 5.3. Let
(
A, G|A

)
be a dense subspace of a cover uniform space

(X, G) and let ϕ :
(
A, G|A

)
→ (Y, H) be a uniformly continuous map into a

complete T2 cover uniform space (Y, H). Then ϕ has unique continuous exten-
sion ϕ̃ : (X, TG) → (Y, TH) and this unique extension is uniformly continuous
as a map from (X, G) to (Y, H).

Theorem 5.4. Every Hausdorff cover uniform space (X, G) has a cover comple-

tion
(
X̃, G̃

)
and every other cover completion (Y, H) is unimorphic to

(
X̃, G̃

)
.

Theorem 5.5. A cover uniform space (X, G) is totally bounded iff every ultra-
filter F on X is Cauchy.

Theorem 5.6. Let
(
A, G|A

)
be a dense subspace of a cover uniform space

(X, G). Then
(
A, G|A

)
is totally bounded iff (X, G) is totally bounded. Also,

every subspace of a totally bounded cover uniform space is totally bounded.

Since every adherence point of an ultrafilter in a topological space is a con-
vergence point and since a topological space is compact iff every ultrafilter
converges, we obtain as a corollary the result of a A. Weil mentioned in the
introduction:



234 A. Garćıa-Máynez

Theorem 5.7. The topology TG of a cover uniform space (X, G) is compact iff
(X, G) is complete and totally bounded.

Normal Wallman bases of Tychonoff spaces determine totally bounded cover
uniform spaces:

Theorem 5.8 ([3]). Let B be a normal Wallman basis of a Tychonoff space
(X, T ). Then the family U (B) of finite covers of X consisting of elements
of B is a totally bounded cover uniformity basis on X and T = TU(B). The

cover completion
(
X̃, Ũ (B)

)
is a Hausdorff compactification of (X, T ) which is

equivalent to the Wallman compactification
(
X (B) , T ∗

)
.

Theorem 5.9 ([3]). Let X, Y be a Tychonoff spaces with respective normal
Wallman bases BX, BY. A map ϕ : (X, U (BX)) → (Y, U (BY)) is uniformly
continuous iff ϕ satisfies the following condition:

∗) whenever H, K ∈ C (BY) are disjoint, there exist disjoint cobasic sets
H1, K1 ∈ C (BX) such that ϕ

−1 (H) ⊆ H1 and ϕ
−1 (K) ⊆ K1.

As a corollary of this last theorem, we obtain the well known universal
property of the Stone–Čech compactification of a Tychonoff space X:

Theorem 5.10. Let ϕ : X → Y be a continuous map from a Tychonoff space
X into a compact Hausdorff space Y. Then ϕ has a continuous extension
ϕ1 : β X → Y.

Proof. Apply last theorem taking as BX, BY the cozero bases of X, Y, respec-
tively. Apply then the extension theorem of uniformly continuous maps into
complete spaces. �

Take a look of [3] for further applications of these theorems.

6. The Wallman type compactification problem

Every annular basis B of a compact Hausdorff space Z is a normal Wallman
basis of Z. If X is a dense subspace of Z, then B|X = {B ∩ X | B ∈ B} is a
Wallman basis of X but B|X may not be normal. Assuming B|X is normal, we
wonder under what conditions Z and X (B|X) are equivalent compactifications
of X. The answer is quite simple:

Z and X (B|X) are equivalent compactifications of X iff Z is the only member
of B containing X (see [13]). If this happens, we say then that B has the trace
property respect to X.

If X is Gδ dense in Z, then the cozero basis of Z has the trace property
respect to X. If B consists of open domains (i.e., sets which coincide with the
interior of their closures) then we have a better result: Z has the trace property
respect to any dense subspace of Z. This suggests the following definition:



Wallman Bases 235

Definition 6.1. A compact T2–space Z is regular Wallman if Z is a Wallman
type compactification of each of its dense subspaces.

As we saw before, if the compact Hausdorff space Z has an annular basis con-
sisting of open domains, then Z is regular Wallman. This happens in compact
metric spaces and, more generally, in arbitrary products of compact metriz-
able spaces (see [14]). Even better, the Stone–Čech compactification of any
metrizable space has an annular basis consisting of open domains and hence
every metrizable space has a normal Wallman basis consisting of open domains.
(See [7]).

Consider the following two questions:

Q1 . Is every compact T2–space regular Wallman?
Q2 . Is every Hausdorff compactification of a Tychonoff space X of Wallman

type?

R. C. Solomon ([11]) answered in the negative the first question proving that

some closed subspace of the cube Ik, where k = (2c)
+
, is not regular Wallman.

As far as the second question is concerned, A. K. Steiner and E. F. Steiner
reduced Q2 to an equivalent problem ([15], 1977):

Q′2 . Is every T2–compactification of a discrete space X of Wallman type?

In the same year (1977), ([16]) Ul’janov answered Q2 in the negative exhibit-
ing a compactification of the discrete space with 2c elements which is not of
Wallman type. He also proved that the continuum hypothesis is equivalent to
the fact that every Hausdorff compactifications of the set of natural numbers
is of Wallman type.

Observe that if a Tychonoff space X has a normal Wallman basis B consist-
ing of open domains, then B∗ is an annular basis of X (B) consisting of open
domains and hence X (B) is regular Wallman.

We finish this section with two questions which, as far as I know, still remain
open:

Q3 . Is every Hausdorff Wallman type compactification of a metrizable space
regular Wallman?

Q4 . Is every normal Wallman basis of a Tychonoff space X equivalent to a
subfamily of the cozero basis of X?

7. A topological dream

I conclude this brief survey on Wallman bases with a topological dream:

“If P is any topological property, there exists a list of conditions
on annular basis of a space X which are equivalent to property
P”.

We have some examples where this dream becomes true:

1) A topological space X is compact and pseudo–metrizable iff X has a
countable normal Wallman basis such that every cobasic set is pseudo-
compact.



236 A. Garćıa-Máynez

2) A Hausdorff space X is compact iff X has an annular basis such that
every cobasic set is compact.

3) A Hausdorff space X is locally compact iff X has a normal Wallman
basis such that every basic set has compact closure or compact com-
plement.

4) A Tychonoff space X is almost compact (i.e., it admits only one com-
patible uniformity) iff X has a normal Wallman basis B such that in
every pair of disjoint cobasic sets, at least one them is compact. By a
previous remark, a Tychonoff space X is almost compact iff it admits
only one (except for equivalence) normal Wallman basis.

We have previously characterized S–metrizable and weakly pseudo–compact
spaces specifying the existence of a normal Wallman basis with some properties.

Consider the following long list of topological properties of Tychonoff spaces.
(I do not include any definition):

compact
metrizable

-
Eberlein
compact

- compact -
locally

compact
Lindelöf

?

locally
compact

paracompact
realcompact

�

locally
compact

realcompact

�

locally
compact

topologically
complete

�
locally

compact

?

almost
locally

compact

-
ultra–

complete
- Čech

complete
- p - countable

type

?

point
countable

type

�k

Can you characterize each of these properties in a similar way?



Wallman Bases 237

References

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[2] J. Dujundji, Topology, Allyn and Bacon, Inc., Boston, 1966.
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[12] L. A. Steen and J. A. Jr. Seebach, Counterexamples in Topology, Springer-Verlag, New
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[13] E. F. Steiner, Wallman spaces and compactifications, Fundamenta Mathematicae 61
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[14] A. K. Steiner and E. F. Steiner, Products of compact metric spaces are regular Wallman,
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[15] A. K. Steiner and E. F. Steiner, On the reduction of the Wallman compactification
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[16] V. M. Ul’janov, Solution of the fundamental problem of bicompact extensions of Wall-
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[17] H. Wallman, Lattices and topological spaces, Ann. of Math. 39 (1938), 112–126.
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Received March 2006

Accepted February 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