Applied General Topology

c© Universidad Politécnica de Valencia

Volume 3, No. 1, 2002

pp. 1–11

Hausdorff compactifications and zero-one
measures II

Georgi D. Dimov
∗

and Gino Tironi
†

Abstract. The notion of PBS-sublattice is introduced and, using
it, a simplification of the results of [6] and of some results of [5] is ob-
tained. Two propositions concerning Wallman-type compactifications
are presented as well.

2000 AMS Classification: Primary 54D35, 28A60, 06D99; Secondary 54D80,
54E05, 06B99.

Keywords: Hausdorff compactifications, zero-one measures on Boolean
algebras, maximal spectrum of distributive lattices, Efremovič proximities,
Wallman-type compactifications.

1. Introduction

In 1977, V. M. Ul′janov ([15]) obtained a negative answer to the famous
Frink’s question, posed in [8], whether each Hausdorff compactification of a
Tychonoff space X is a Wallman-type compactification (we shall use from now
on the term “Wallman compactification” instead of “Wallman-type compact-
ification”). O. Frink introduced the Wallman compactifications of a space X
as spaces of all C-ultrafilters, where C is a ring of subsets of X and a special
closed base of X (called normal base) (we will denote such compactifications
by ω(X,C)). Passing to the complements in X of all elements of a normal base
C, one obtains a special open base B = C′ of X (which is again a ring of sets),
called normal Wallman base. This leads to a dual description of the Wallman
compactifications of X as spaces of the type max(B) (= maximal spectrum of
B), where B is a normal Wallman base of X (see, e.g., [9]). Hence, in general,
not every Hausdorff compactification of a Tychonoff space X can be obtained
as a maximal spectrum of a normal Wallman base of X. In our paper [6], using

∗The first author was partially supported by a Fellowship for Mathematics of the NATO-

CNR Outreach Fellowships Programme 1999, Bando 219.32/16.07.1999.
†The second author was supported by the National Group “Analisi reale” of the Italian

Ministry of the University and Scientific Research at the University of Trieste.



2 G. D. Dimov and G. Tironi

the notion of PB-sublattice introduced in [5], we answered affirmatively two
natural questions. The first one was:

Problem 1.1. Is it possible to correlate (in a canonical way) to each Tychonoff
space X a Boolean algebra BX and a set LX of sublattices of BX in order to
obtain that the set of all, up to equivalence, Hausdorff compactifications of X
is represented by the set {max(L) : L ∈LX}?

This question was motivated also by some measure-theoretic constructions
of Hausdorff compactifications. It was well known (see [1, 3, 4, 14]) that, when
C is a normal base of X, then the space IR(C) (of all regular zero-one measures
on the Boolean subalgebra b(C) of the Boolean algebra exp(X) (of all subsets
of X, with the natural operations), generated by the sublattice C of exp(X))
is a Hausdorff compactification of X equivalent to ω(X,C) and max(C′). The
second problem was:

Problem 1.2. Is it possible to construct in a similar way (by means of zero-one
measures) every Hausdorff compactification of X?

In this paper we introduce the notion of PBS-sublattice and, using it, we
obtain a simplification of the results of [6] and of some results of [5]. We also
present the notion of PB-sublattice in a simpler but equivalent form. Finally,
a necessary and sufficient condition and a sufficient condition, as well, which
a lattice L ∈ LX has to satisfy in order to obtain that max(L) is a Wallman
compactification of X, are stated and proved.

2. Preliminaries

We first fix some notations.

Note 2.1. We denote by ω the set of all positive natural numbers. All lattices
will be with top (= unit) and bottom (= zero) elements, denoted respectively
by 1 and 0 and all sublattices of a lattice L are assumed to contain the top
and the bottom elements of L. We don’t require the elements 0 and 1 to be
distinct.

Let A be a distributive lattice. The set of all ideals of A will be denoted by
Idl(A) and the set of all maximal ideals of A (which will be, as usual, always
proper) — by max(A). Put TA = {OI = {J ∈ max(A) : I 6⊆ J} : I ∈ Idl(A)}.
The space (max(A),TA) is called maximal spectrum of A and the topology TA
is called spectral topology on the set max(A). (max(A),TA) is always a compact
T1-space (see, e.g., [9]). If the lattice A is normal (i.e., for each pair a,b ∈ A
with a∨ b = 1, there exist u,v ∈ A such that a∨u = 1 = b∨v and u∧v = 0)
then (max(A),TA) is a compact T2-space.

If L is a sublattice of a Boolean algebra B then we will denote by b(L) the
Boolean subalgebra of B generated by L. By exp(X) we denote the set of all
subsets of the set X.

The ordered set of all, up to equivalence, Hausdorff compactifications of a
Tychonoff space X will be denoted by (K(X),≤).



Hausdorff compactifications and zero-one measures II 3

If (X,T ) is a topological space then we write Coz(X,T ) or, simply, Coz(X)
for the set of all cozero-subsets of X; the closure of a subset M of (X,T ) will
be denoted by clX M; a dense embedding will mean an embedding with dense
image.

By a proximity we shall always mean an Efremovič proximity. If δ is a
proximity on a set X, then δ will be the complement of the relation δ. If
(X,T ) is a topological space and δ is a proximity on the set X, we say that
δ is a proximity on the space (X,T ) if the topology Tδ, generated by δ on the
set X, coincides with T . The ordered set of all proximities δ on a topological
space (X,T ) will be denoted by (PT (X),≤).

For all undefined terms and notations see [7], [9] and [11].

We shall recall the Smirnoff Compactification Theorem:

Theorem 2.2 ([13]). Let (X,T ) be a Tychonoff space. If (cX,c) is a Hausdorff
compactification of X, then putting, for every A,B ⊆ X, AδcB iff clcX c(A) ∩
clcX c(B) = ∅, we obtain a proximity δc on (X,T ). The correspondence

s: (K(X),≤) −→ (PT (X),≤),
defined by s(cX,c) = δc, is an isomorphism.

If δ ∈ PT (X) then the compactification s−1(δ) of X, which will be denoted
by (cδX,cδ), is called Smirnoff compactification of (X,T ).

Definition 2.3 ([9, 8]). Let (X,T ) be a topological space. A sublattice B of
T is called a Wallman base for X if B is a base of T and satisfies the following
condition:

(W) Whenever U ∈B and x ∈ U, there exists V ∈B with U ∪V = X and
x 6∈ V .

If B is a Wallman base for a T0-space X, then the map
ηB : X −→ max(B), x 7→ ηB(x) = {U ∈B : x 6∈ U},

is a dense embedding. Hence, for every T1-space X, (max(B),ηB) is a T1-
compactification of X. If B is a normal Wallman base, then (max(B),ηB) is a
T2-compactification of X, called Wallman compactification.

A family C of closed subsets of X, such that the family B = C′ = {X \F :
F ∈ C} is a normal Wallman base of X, is called a normal base of X. Let
ω(X,C) denote the set of all C-ultrafilters. Topologize this set by using as a
base for the closed sets all sets of the form A− = {F ∈ ω(X,C) : A ∈ F},
where A ∈ C. Then the map ωC : X −→ ω(X,C), defined by the formula
ωC(x) = {F ∈C : x ∈ F}, where x ∈ X, is a dense embedding of X in ω(X,C)
and (ω(X,C),ωC) is a compactification of X equivalent to (max(B),ηB).

We will need the following theorem of O. Nj̊astad:

Theorem 2.4 ([12]). Let (X,T ) be a Tychonoff space. A compactification
(cX,c) of X is a Wallman compactification if and only if there exists a sub-
family B of T which is closed under finite unions and satisfies the following
two conditions:



4 G. D. Dimov and G. Tironi

(B1) If U,V ∈B and U ∪V = X then (X \U)δc(X \V );
(B2) If A,B ⊆ X and AδcB then there exist U,V ∈B such that A ⊆ X \U,

B ⊆ X \V and U ∪V = X.
Recall that (see, e.g., [1, 3, 4, 14]) a measure on a Boolean algebra A is a

non-negative real-valued function µ on A such that µ(a∨ b) = µ(a) + µ(b) for
all a,b ∈ A with a∧b = 0; in the case when µ(A) = {0, 1}, µ is called a zero-one
measure.

Let B be a Boolean algebra and L be a sublattice of B. A measure µ, defined
on the Boolean algebra b(L), is called L-regular measure (or, simply, regular
measure) if µ(x) = sup{µ(a) : a ∈ L,x ≥ a} for any x ∈ b(L). The set of all
L-regular zero-one measures on the Boolean algebra b(L) will be denoted by
IR(L). The topology Dw on IR(L) is defined as follows: a base for the closed
sets of Dw consists of all sets of the form W(a) = {µ ∈ IR(L) : µ(a) = 1},
where a ∈ L. The space (IR(L),Dw) is a compact T1-space.

If X is a Tychonoff space and C is a normal base of X then (IR(C),Dw) is
a compact Hausdorff space. The map MC : X −→ (IR(C),Dw), defined by the
formula MC(x) = µx, where x ∈ X and, for every element F of the Boolean
subalgebra b(C) of exp(X),

µx(F) = 1 if x ∈ F, and mux(F) = 0 if x 6∈ F,
is a dense embedding. ((IR(C),Dw),MC) is a compactification of X equivalent
to (ω(X,C),ωC) and (max(C′),ηC′).

We will recall a theorem of J. Kerstan.

Definition 2.5 ([10, 2]). A family U of open subsets of a topological space
space X is called completely regular if for every U ∈U there exist two sequences
(Ui)i∈ω and (V i)i∈ω in U such that U =

⋃
{Ui : i ∈ ω} and Ui ⊆ X \V i ⊆ U

for each i ∈ ω.
Theorem 2.6 ([10, 2]). A subset of a topological space is a cozero-set if and
only if it belongs to a completely regular family.

3. The Results

Definition 3.1. Let (X,T ) be a space and U be an open subset of X. If there
is a sequence (Ui,Uci)i∈ω in T ×T with U =

⋃
i∈ω U

i, Ui ⊆ X\Uci ⊆ Ui+1, for
every i ∈ ω, then such a sequence (Ui,Uci)i∈ω will be called Ur−representation
of U. We put TUr = {U ∈T : U has an Ur-representation}.
Definition 3.2. Let (X,T ) be a space. Denote by L(X) the set of all Ur-re-
presentations of the elements of TUr. The elements of L(X) will be written in
the following way:

Ū = (Ui,Uci)i∈ω,
where (Ui,Uci)i∈ω is a Ur-representation of U0 =

⋃
{Ui : i ∈ ω}; two elements

Ū = (Ui,Uci)i∈ω and V̄ = (V i,V ci)i∈ω of L(X) are equal if Ui = V i, Uci =
V ci, for every i ∈ ω. Define two operations ∧ and ∨ in L(X) by

Ū ∨ V̄ = (Ui ∪V i,Uci ∩V ci)i∈ω



Hausdorff compactifications and zero-one measures II 5

and
Ū ∧ V̄ = (Ui ∩V i,Uci ∪V ci)i∈ω,

where Ū = (Ui,Uci)i∈ω and V̄ = (V i,V ci)i∈ω, and let 0̄ = (0i, 0ci)i∈ω, 1̄ =
(1i, 1ci)i∈ω, where ∅ = 0i = 1ci, X = 1i = 0ci, i ∈ ω.

Fact 3.3. (L(X),∨,∧) is a distributive lattice and 0̄, 1̄ are its zero and one.

Definition 3.4 (see also [5]). Let X be a Tychonoff space. A sublattice L of
L(X) is said to be a PB-sublattice if

(L1) The set L0 = {U0 =
⋃
{Ui : i ∈ ω} : (Ui,Uci)i∈ω ∈ L} is an open base

of the space X;
(L2) For every Ū = (Ui,Uci)i∈ω ∈ L and every j ∈ ω, there exist k ∈ ω and

V̄ = (V i,V ci)i∈ω,W̄ = (Wi,Wci)i∈ω ∈ L (which depend on the choice
of Ū and j) such that Uc(j+1) ⊆ Wk ⊆ W 0 = Ucj, Uj−1 ⊆ V k ⊆ V 0 =
Uj (for j > 1), and V 0 = Uj (for j = 1).

Proposition 3.5. Let L be a PB-sublattice of L(X). Then, for every element
Ū = (Ui,Uci)i∈ω of L and for every i ∈ ω, we have that Ui,Uci ∈ Coz(X).
Hence, L0 ⊆ Coz(X).

Proof. For every Ū = (Ui,Uci)i∈ω ∈ L and every j ∈ ω, we have, by (L2),
that there exist V̄ = (V i,V ci)i∈ω ∈ L and W̄ = (Wi,Wci)i∈ω ∈ L such that
Uj = V 0 and Ucj = W 0. Hence, in order to prove our proposition, we need
only to show, according to Kerstan Theorem (see 2.6), that L0 is a completely
regular family (see 2.5). So, let Ū = (Ui,Uci)i∈ω ∈ L. Then {Ui : i ∈ ω}⊆ L0
and U0 =

⋃
{Ui : i ∈ ω}. We let (Ui)i∈ω to be the first required sequence.

As it follows from 3.1, (Uci)i∈ω can serve as the second required sequence.
Therefore, L0 is a completely regular family. �

Definition 3.6 ([5]). Let (X,τ) be a space. Denote by L(Coz(X)) the set of
all Ur-representations of all elements of Coz(X) by elements of Coz(X). We
will regard L(Coz(X)) as a sublattice of the lattice L(X).

Remark 3.7. Let us remark that in [5] the notion of “PB-sublattice” was
introduced with the redundant (as Proposition 3.5 shows now) requirement
that a PB-sublattice is (by definition) a sublattice of L(Coz(X)).

Proposition 3.8 ([5]). (L(Coz(X)),∨,∧) is the greatest (with respect to the
inclusion) PB-sublattice of (L(X),∨,∧).

Note 3.9. Let X be a set. We will denote by S(X) the complete Boolean
algebra (exp(X))ℵ0 .

Definition 3.10. Let (X,T ) be a topological space. We put
OIS(X,T ) = {Ū = (Ui)i∈ω : Ui ∈T ,Ui ⊆ Ui+1,∀i ∈ ω}.

Instead of OIS(X,T ), we shall often write simply OIS(X). For every (Ui)i∈ω ∈
OIS(X), we put U0 =

⋃
{Ui : i ∈ ω}. We will regard OIS(X) as a sublattice

of S(X).



6 G. D. Dimov and G. Tironi

Definition 3.11. Define a relation ∼ in S(X) putting: for every Ū = (Ui)i∈ω,
V̄ = (V i)i∈ω ∈ OIS(X), Ū ∼ V̄ if and only if there exists an i0 ∈ ω such that
Ui = V i, for every i ≥ i0. Then ∼ is a congruence relation on the Boolean
algebra S(X). So, a quotient Boolean algebra S(X)/∼, which will be denoted
by [S(X)], is defined. The natural mapping between S(X) and [S(X)] will be
denoted by π. We put, for every Ū ∈ S(X), π(Ū) = [Ū].

Definition 3.12. Let (X,T ) be a Tychonoff space. A sublattice L of the
lattice OIS(X) is said to be a PBS-sublattice in X, if
(LS1) The set L0 = {U0 : (Ui)i∈ω ∈ L} is a base of the space X;
(LS2) For every Ū = (Ui)i∈ω ∈ L and for every j ∈ ω, there exist V̄ =

(V i)i∈ω,W̄ = (Wi)i∈ω ∈ L and k ∈ ω (which depend on the choice of
Ū and j) such that X\Uj+1 ⊆ V k ⊆ V 0 ⊆ X\Uj, and Uj−1 ⊆ Wk ⊆
W 0 = Uj (for j > 1), Uj = W 0 (for j = 1).

Fact 3.13. The restriction of the relation ∼ (defined in 3.11) to any PBS-
sublattice L in X is a congruence relation in L. So, a quotient lattice [L] = L/∼
is defined.

Lemma 3.14. Let L′ be a PB-sublattice of L(X). Then

L = {Ū = (Ui)i∈ω : there exists an Ū′ ∈ L′ such that Ū′ = (Ui,Uci)i∈ω}

is a PBS-sublattice in X. For every Ū′ = (Ui,Uci)i∈ω ∈ L′ put p(Ū′) =
(Ui)i∈ω. Then p: L′ −→ L is a lattice homomorphism, L = p(L′) and the
correspondence

[p] : [L′] −→ [L], [Ū′] −→ [p(Ū′)]
is a lattice isomorphism.

Proof. For proving that L is a PBS-sublattice in X, we need only to check that
the first part in the condition (LS2) (see 3.12) is satisfied.

Let Ū = (Ui)i∈ω ∈ L and j ∈ ω. There exists an Ū′ ∈ L′ such that Ū′ =
(Ui,Uci)i∈ω. By (L2) of 3.4, there exist W̄ ′ = (Wi,Wci)i∈ω ∈ L′ and l ∈ ω
such that Uj ⊆ W l ⊆ W 0 = Uj+1. Then Uj ⊆ W l ⊆ W l+2 ⊆ Uj+1 and hence
X\Uj ⊇ X\W l ⊇ X\W l+2 ⊇ X\Uj+1. We have that X\W l+2 ⊆ Wc(l+1) ⊆
X\W l+1 ⊆ Wcl ⊆ X\W l. Using again (L2) of 3.4, we obtain that there exist
V̄ ′ = (V i,V ci)i∈ω ∈ L′ and k ∈ ω such that Wc(l+1) ⊆ V k ⊆ V 0 = Wcl.
Therefore V̄ = (V i)i∈ω ∈ L and X \Uj+1 ⊆ V k ⊆ V 0 ⊆ X \Uj.

It is easy to see that [p] is a lattice isomorphism. �

Lemma 3.15. For every PBS-sublattice L in X there exists a PB-sublattice
L′ of L(X) such that p(L′) = L and [p] : [L′] −→ [L] is a lattice isomorphism
(see 3.14 for the notations).

Proof. Let Ū = (Ui)i∈ω ∈ L. Then, by (LS2) (see 3.12), for every j ∈ ω there
exist V̄j = (V ij )i∈ω ∈ L and kj ∈ ω such that U

j ⊆ X \V 0j ⊆ X \V
kj
j ⊆ U

j+1.
Put Ucj = V 0j , for every j ∈ ω. Then U

j ⊆ X \ Ucj ⊆ Uj+1, for every
j ∈ ω, and hence Ū′ = (Ui,Uci)i∈ω ∈ L(X). Put L′′ = {Ū′ : Ū ∈ L}. Then



Hausdorff compactifications and zero-one measures II 7

L′′ ⊆ L(X). Let L′ be the sublattice of L(X) generated by L′′. In order to
show that L′ is a PB-sublattice of L(X), we need only to check that the first
part of the condition (L2) (see 3.4) is satisfied. Let Ū = (Ui)i∈ω ∈ L. Then
Ū′ = (Ui,Uci)i∈ω ∈ L′′. Let j ∈ ω. By the construction of Ū′, we have that
Uc(j+1) ⊆ X \Uj+1 ⊆ V kjj ⊆ V

0
j = U

cj. Since (V̄j)′ ∈ L′′ ⊆ L′ and kj ∈ ω, we
obtain that (L2) is satisfied by the elements of L′′. From the facts that L is a
lattice and L′′ generates L′, we obtain that (L2) is satisfied also by all elements
of L′. So, L′ is a PB-sublattice of L(X). The construction of L′ shows that
p(L′) = L. The rest follows from 3.14. �

Corollary 3.16. Let L be a PBS-sublattice in (X,T ). Then, for every element
Ū = (Ui)i∈ω of L and for every i ∈ ω, we have that Ui ∈ Coz(X). Hence,
L0 ⊆ Coz(X).

Proof. It follows from 3.15 and 3.5. �

Theorem 3.17. Let (X,T ) be a Tychonoff space and L be a PBS-sublattice
in X. Define for A,B ⊆ X: AδLB iff there exist Ū = (Ui)i∈ω ∈ L and k ∈ ω
such that A ⊆ Uk ⊆ U0 ⊆ X \B. Then δL is an Efremovič proximity on the
topological space (X,T ). (We will say that the proximity δL is generated by
the PBS-sublattice L in X.) Moreover, for any proximity δ on (X,T ) there
exists a PBS-sublattice L in X such that δ = δL. The set of all PBS-sublattices
in X generating a proximity δ on (X,T ) has a greatest element (with respect
to the inclusion), which will be denoted by Lδ.

Proof. By Lemma 3.15, there exists a PB-sublattice L′ of L(X) such that
p(L′) = L. In Proposition 2.12 of [5] we show that the relation δL′ generated
by L′, defined in the same way as we define here the relation δL, is a proximity
on the space (X,T ). Hence, δL is such one, as well. This fact can be also
obtained directly, modifying the proof of Proposition 2.12 of [5].

If (cX,c) is a compactification of X then the family F = {f : X −→ [0, 1] : f
has a continuous extension to cX} generates (cX,c). The PB-sublattice LF
of L(X), constructed in Example 2.4 of [5], has the property that δLF = δc
(see Theorem 3.1(a) of [5]). By Lemma 3.14, the lattice L = p(LF) is a
PBS-sublattice in X. Obviously, δLF = δL. Hence, by Theorem 2.2, for any
proximity δ on (X,T ) there exists a PBS-sublattice L in X such that δ = δL.

Finally, one easily infer from Proposition 2.11 of [5] and our lemmas 3.14
and 3.15 that the set of all PBS-sublattices in X generating a proximity δ on
(X,T ) has a greatest element (with respect to the inclusion). �

Theorem 3.18. Let (X,T ) be a Tychonoff space and L be a PBS-sublattice in
X. Put, for every x ∈ X, Ix = {Ū ∈ L : x 6∈ U0}. Then:

(a) π(Ix) = {[Ū] : Ū ∈ Ix} ∈ max([L]) and the map eL : (X,T ) −→
max([L]), defined by the formula eL(x) = π(Ix), is a dense embedding;

(b) (max([L]),eL) is a Hausdorff compactification of X, equivalent to the
Smirnoff compactification (cδLX,cδL) (see 3.17 for δL and 2.2 for
(cδLX,cδL)).



8 G. D. Dimov and G. Tironi

Hence, the set K(X) of all, up to equivalence, Hausdorff compactifications of
X is represented by the set {(max([Lδ]),eLδ ) : δ ∈ PT (X)}. Moreover, the
following is true: (cδ1X,cδ1 ) ≤ (cδ2X,cδ2 ) iff Lδ1 ⊆ Lδ2 (see 3.17 for Lδ).

Therefore, putting BX = [S(X)] and LX = {[Lδ] : δ ∈ PT (X)}, we obtain a
new (simpler) solution of our Problem 1.

Proof. In [6] the PB-sublattice version of this theorem (i.e., the version obtained
by substituting everywhere in the theorem “PBS-” with “PB-”) was proved (see
Theorem 3.8 there). Now our result follows from it, from lemmas 3.14, 3.15
and Theorem 3.17 proved above, and from 2.17, 2.13 of [5]. �

Definition 3.19. Let B be a Boolean algebra and L be a sublattice of B. A
measure µ, defined on the Boolean algebra b(L), is called u-regular measure (or
u-L-regular measure) if µ(x) = inf{µ(a) : a ∈ L,x ≤ a} for any x ∈ b(L). The
set of all u-L-regular zero-one measures on the Boolean algebra b(L) will be
denoted by Iur(L).

The following lemma is essentialy known (see [1], Theorem 2.1):

Lemma 3.20. Let B be a Boolean algebra and L be a sublattice of B. Then
there exists a bijection between the sets max(L) and Iur(L).

Lemma 3.21. Let (X,T ) be a Tychonoff space and L be a PBS-sublattice in
X. Then [L] is a sublattice of [S(X)] (see 3.11 and 3.9 for the notations). For
every [Ū] ∈ [L] put

[Ū]∗ = {µ ∈ Iur([L]) : µ([Ū]) = 1}.
Then the family B∗ = {[Ū]∗ : [Ū] ∈ [L]} is a base of a topology T ∗ on the set
Iur([L]). If δ is the proximity on (X,T ) generated by L (see 3.17), then for
every x ∈ X and every [Ū] = [(Ui)i∈ω] ∈ b([L]) put:

µx([Ū]) =

{
0 if there exists an i0 ∈ ω such that xδUi for every i ≥ i0, and
1 if there exists an j0 ∈ ω such that xδ(X \Uj) for every j ≥ j0.

Then, for every x ∈ X, µx is a well-defined zero-one u-[L]-regular measure on
the Boolean subalgebra b([L]) of the complete Boolean algebra [S(X)] and the
mapping mL : (X,T ) −→ (Iur([L]),T ∗), defined by the formula mL(x) = µx,
is a dense embedding. ((Iur([L]),T ∗),mL) is a Hausdorff compactification of
(X,T ) equivalent to the compactification (max([L]),eL) of (X,T ) (and, hence,
to the Smirnoff compactification (cδX,cδ)). The map Φ : (Iur([L]),T ∗) −→
max([L]), defined by the formula Φ(µ) = µ−1(0) ∩ [L], carries out this equiva-
lence.

Proof. In [6] the PB-sublattice version of this lemma was proved (see Lemma
3.16 there). Our result follows from it and from Lemma 3.15 proved above. �

Theorem 3.22 (The Main Theorem). Let (X,T ) be a Tychonoff space.
Then for every Hausdorff compactification (cX,c) of X there exists a sublat-
tice [L] of the complete Boolean algebra [S(X)] (where L is a PBS-sublattice
in X) such that (max([L]),eL) (see 3.18 for the definition of the map eL) and



Hausdorff compactifications and zero-one measures II 9

((Iur([L]),T ∗),mL) (see 3.19 and 3.21 for the notations) are Hausdorff com-
pactification of X equivalent to the compactification (cX,c) of X.

Proof. Let (cX,c) be a Hausdorff compactification of (X,T ). Then, by Theo-
rem 3.17, there exists a PBS-sublattice L in X such that δL = δc (see 2.2 and
3.17 for the notations). Now, Theorem 3.18, Lemma 3.21 and Theorem 2.2 im-
ply that (max([L]),eL) and ((Iur([L]),T ∗),mL) are Hausdorff compactification
of X equivalent to the compactification (cX,c) of X. �

Corollary 3.23. Let (X,T ) be a Tychonoff space. Put

MA(X) = {(max([Lδ]),eLδ ) : δ ∈ PT (X)}

and
ME(X) = {((Iur([Lδ]),T ∗),mLδ ) : δ ∈ PT (X)}.

Order these sets putting for every δ1,δ2 ∈ PT (X),

max([Lδ1 ]) ≤ max([Lδ2 ]) iff δ1 ≤ δ2, and Iur([Lδ1 ]) ≤ Iur([Lδ2 ]) iff δ1 ≤ δ2.

Then the ordered sets (MA(X),≤) and (ME(X),≤) are isomorphic to the
ordered set (K(X),≤) of all, up to equivalence, Hausdorff compactifications of
X.

In the next proposition, the O. Nj̊astad’s characterization of Wallman com-
pactifications by means of proximities (see 2.4) is restated in the language of
PBS-sublattices.

Proposition 3.24. Let (X,T ) be a Tychonoff space and L be a PBS-sublattice
in X. Then (max([L]),eL) is a Wallman compactification of X iff there exists
a family B, consisting of open subsets of X, such that

(i) B is closed under finite unions;
(ii) If A,B ∈ B and A ∪ B = X then there exist Ū = (Ui)i∈ω ∈ L and

j ∈ ω with X \A ⊆ Uj ⊆ U0 ⊆ B;
(iii) If Ū = (Ui)i∈ω ∈ L and j ∈ ω then there exist A,B ∈ B such that

Uj ⊆ X \A ⊆ B ⊆ U0.

Proof. The proximity generated by the compactification (max([L]),eL) is
exactly the proximity δL (see Theorem 3.18(b)). Hence, by Theorem 2.4,
(max([L]),eL) is a Wallman compactification of X if and only if there exists a
subfamily B of T which is closed under finite unions and satisfies the conditions
(B1) and (B2). Since our proximity δL is generated by L, these conditions can
be rewritten now as follows:

(B1L) If U,V ∈ B and U ∪ V = X then there exist Ū = (Ui)i∈ω ∈ L and
j ∈ ω such that X \U ⊆ Uj ⊆ U0 ⊆ V ;

(B2L) If A,B ⊆ X and there exist Ū = (Ui)i∈ω ∈ L and j ∈ ω such that
A ⊆ Uj ⊆ U0 ⊆ X \ B then there are V,W ∈ B with A ⊆ X \ V ,
B ⊆ X \W and V ∪W = X.



10 G. D. Dimov and G. Tironi

Obviously, condition (B1L) coincides with condition (ii) of our Proposition
and condition (i) is also satisfied. Since for every Ū = (Ui)i∈ω ∈ L and j ∈ ω
we have that UjδL(X \ U0), condition (B2L) is equivalent to the condition
(iii). This completes the proof. �

Now we will give a sufficient condition for (max([L]),eL) to be a Wallman
compactification:

Proposition 3.25. Let (X,T ) be a Tychonoff space and L be a PBS-sublattice
in X. If L satisfies the following condition:

(Wa) If Ū, V̄ ∈ L and U0 ∪V 0 = X then there exist W̄ = (Wi)i∈ω ∈ L and
j ∈ ω such that X \U0 ⊆ Wj ⊆ W 0 ⊆ V 0,

then (max([L]),eL) is a Wallman compactification of X. In fact, we have that
(max([L]),eL) is equivalent to the Wallman compactification (max(L0),ηL0 )
(see 2.3 and 3.12 for the notations).

Proof. Let us recall that O. Nj̊astad ([12]) proved that if (X,δ) is a proximity
space then a subfamily B of the topology Tδ, generated by the proximity δ, is a
normal Wallman base of (X,Tδ) if it is a ring of sets and satisfies the conditions
(B1) and (B2) from 2.4; moreover, he showed that (max(B),ηB) and (cδX,cδ)
are equivalent compactifications of (X,Tδ) (see 2.2 and 2.3 for the notations).

By Theorem 3.18(b), we have that (max([L]),eL) and (cδLX,cδL) are equiv-
alent compactifications of (X,T ). Obviously, L0(= {U0 =

⋃
{Ui : i ∈ ω} :

(Ui)i∈ω ∈ L}) is a ring of open sets in (X,T ) and T = TδL (see Theo-
rem 3.17). So, in order to prove our proposition, it is enough to show that
the family L0 satisfies the conditions (B1) and (B2) from 2.4. The condition
(Wa) says that if U0,V 0 ∈ L0 and U0 ∪ V 0 = X then (X \ U0)δL(X \ V 0).
Hence (B1) is satisfied. For proving (B2), let A,B ⊆ X and AδLB. Then,
by the definition of δL, there exist Ū = (Ui)i∈ω ∈ L and j ∈ ω such that
A ⊆ Uj ⊆ U0 ⊆ X \ B. Since L is a PBS-sublattice in X, we obtain (by
the condition (LS2) from Definition 3.12) that there exist V̄ = (V i)i∈ω ∈ L
and k ∈ ω with Uj ⊆ X \ V 0 ⊆ X \ V k ⊆ Uj+1 ⊆ U0. Hence A ⊆ X \ V 0,
B ⊆ X \U0, U0 ∪V 0 = X and U0,V 0 ∈ L0. Therefore, L0 satisfies (B2). The
proof of our proposition is completed. �

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Hausdorff compactifications and zero-one measures II 11

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Received February 2001

Revised May 2001

Georgi D. Dimov

Department of Mathematics and Computer Science
University of Sofia
Blvd. J. Bourchier 5
1126 Sofia, Bulgaria

E-mail address : gdimov@fmi.uni-sofia.bg

Gino Tironi

Department of Mathematical Sciences
University of Trieste
Via A. Valerio 12/1
34127 Trieste, Italy

E-mail address : tironi@univ.trieste.it