BouMikuRichAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 10, No. 2, 2009

pp. 173-186

Convergence semigroup actions: generalized
quotients

H. Boustique, P. Mikusiński and G. Richardson

Abstract. Continuous actions of a convergence semigroup are in-
vestigated in the category of convergence spaces. Invariance properties

of actions as well as properties of a generalized quotient space are pre-

sented

2000 AMS Classification: 54A20, 54B15.

Keywords: Continuous action, convergence space, quotient map, semigroup.

1. Introduction

The notion of a topological group acting continuously on a topological space
has been the subject of numerous research articles. Park [8, 9] and Rath [10]
studied these concepts in the larger category of convergence spaces. This is a
more natural category to work in since the homeomorphism group on a space
can be equipped with a coarsest convergence structure making the group oper-
ations continuous. Moreover, unlike in the topological context, quotient maps
are productive in the category of all convergence spaces with continuous maps
as morphisms. This property plays a key role in the proof of several results
contained herein; for example, Theorem 4.11.

Given a topological semigroup acting on a topological space, Burzyk et al.
[1] introduced a ”generalized quotient space.” Elements of this space are equiv-
alence classes determined by an abstraction of the method used to construct
the rationals from the integers. General quotient spaces are used in the study
of generalized functions [5, 6, 7].

Generalized quotients in the category of convergence spaces are studied in
section 4. First, invariance properties of continuous actions of convergence
semigroups on convergence spaces are investigated in section 3.



174 H. Boustique, P. Mikusiński and G. Richardson

2. Preliminaries

Basic definitions and concepts needed in the area of convergence spaces are
given in this section. Let X be a set, 2X the power set of X, and let F(X)
denote the set of all filters on X. Recall that B ⊆ 2X is a base for a filter on
X provided B 6= ∅, ∅ /∈ B, and B1,B2 ∈ B implies that there exists B3 ∈ B
such that B3 ⊆ B1 ∩ B2. Moreover, [B] denotes the filter on X whose base is
B; that is, [B] = {A ⊆ X : B ⊆ A for some B ∈ B}. Fix x ∈ X, define ẋ to
be the filter whose base is B = {{x}}. If f : X → Y and F ∈ F(X), then f→F
denotes the image filter on Y whose base is {f(F) : F ∈ F}.

A convergence structure on X is a function q : F(X) → 2X obeying :
(CS1) x ∈ q(ẋ) for each x ∈ X
(CS2) x ∈ q(F) implies that x ∈ q(G) whenever F ⊆ G.
The pair (X,q) is called a convergence space. The more intuitive notation

F
q
→ x is used for x ∈ q(F). A map f : (X,q) → (Y,p) between two conver-

gence spaces is called continuous whenever F
q
→ x implies that f→F

p
→ f(x).

Let CONV denote the category whose objects consist of all the convergence
spaces, and whose morphisms are all the continuous maps between objects. The
collection of all objects in CONV is denoted by |CONV|. If p and q are two

convergence structures on X, then p ≤ q means that F
p
→ x whenever F

q
→ x.

In this case, p(q) is said to be coarser(finer) than q(p), respectively. Also, for
F,G ∈ F(X), F ≤ G means that F ⊆ G, and F(G) is called coarser(finer)
than G(F), respectively.

It is well-known that CONV possesses initial and final convergence struc-
tures. In particular, if (Xj,qj ) ∈ |CONV| for each j ∈ J, then the prod-

uct convergence structure r on X = ×
j∈J

Xj is given by H
r
→ x = (xj ) iff

π→j H
qj
→ xj for each j ∈ J, where πj denotes the j

th projection map. Also, if
f : (X,q) → Y is a surjection, then the quotient convergence structure

σ on Y is given by H
σ
→ y iff there exists x ∈ f−1(y) and F

q
→ x such that

f→F = H. In this case, σ is the finest convergence structure on Y making
f : (X,q) → (Y,σ) continuous.

Unlike the category of all topological spaces, CONV is cartesian closed and
thus has suitable function spaces. In particular, let (X,q), (Y,p) ∈ |CONV|
and let C(X,Y ) denote the set of all continuous functions from X to Y . Define
ω : (X,q)×C(X,Y ) → (Y,p) to be the evaluation map given by ω(x,f) = f(x).
There exists a coarsest convergence structure c on C(X,Y ) such that w is

jointly continuous. More precisely, c is defined by : Φ
c
→ f iff w→(F × Φ)

p
→

f(x) whenever F
q
→ x. This compatibility between (X,q) and (C(X,Y ),c) is

an example of a continuous action in CONV discussed in section 3. Continuous
actions which are invariant with respect to a convergence space property P are
studied in section 3. Choices for P include : locally compact, locally bounded,
regular, Choquet(pseudotopological), and first-countable.



Convergence semigroup actions: generalized quotients 175

An object (X,q) ∈ |CONV| is said to be locally compact (locally bounded)

if F
q
→ x implies that F contains a compact (bounded) subset of X, respec-

tively. A subset B of X is bounded provided that each ultrafilter contain-
ing B q-converges in X. Further, (X,q) is called regular (Choquet) pro-

vided clqF
q
→ x (F

q
→ x) whenever F

q
→ x (each ultrafilter containing F

q-converges to x), respectively. Here clqF denotes the filter on X whose base
is {clqF : F ∈ F}. Some authors use the term ”pseudotopological space”
for a Choquet space. Finally, (X,q) is said to be first-countable whenever

F
q
→ x implies the existence of a coarser filter on X having a countable base

and q-converging to x.
Let SG denote the category whose objects consist of all the semigroups

(with an identity element), and whose morphisms are all the homomorphisms
between objects. Further, (S,.,p) is said to be a convergence semigroup
provided : (S,.) ∈ |SG|, (S,p) ∈ |CONV|, and γ : (S,p) × (S,p) → (S,p)
is continuous, where γ(x,y) = x.y. Let CSG be the category whose objects
consist of all the convergence semigroups, and whose morphisms are all the
continuous homomorphisms between objects.

3. Continuous Actions

An action of a semigroups on a topological space is used to define ”general-
ized quotients” in [1]. Below is Rath’s [10] definition of an action in the conver-
gence space context. Let (X,q) ∈ |CONV|, (S,.,p) ∈ |CSG|, λ : X × S → X,
and consider the following conditions :
(a1) λ(x,e) = x for each x ∈ X (e is the identity element)
(a2) λ(λ(x,g),h) = λ(x,g.h) for each x ∈ X, g,h ∈ S
(a3) λ : (X,q) × (S,.,p) → (X,q) is continuous.
Then (S,.)((S,.,p)) is said to act(act continuously) on (X,q) whenever a1-
a2 (a1-a3) are satisfied and, in this case, λ is called the action (continuous
action), respectively. For sake of brevity, (X,S) ∈ A(AC) denotes the fact
that (S,.,p) ∈ |CSG|) acts (acts continuously) on (X,q) ∈ |CONV|, respec-
tively. Moreover, (X, S, λ) ∈ A indicates that the action is λ.

The notion of ”generalized quotients” determined by commutative semi-
group acting on a topological space is investigated in [1]. Elements of the
semigroup in [1] are assumed to be injections on the given topological space.

Lemma 3.1 ([1]). Suppose that (S,X,λ) ∈ A, (S,.) is commutative and
λ(.,g) : X → X is an injection, for each g ∈ S. Define (x,g) ∼ (y,h) on
X × S iff λ(x,h) = λ(y,g). Then ∼ is an equivalence relation on X × S.

In the context of Lemma 3.1, let 〈(x,g)〉 be the equivalence class containing
(x,g), B(X, S) denote the quotient set (X×S)/ ∼, and define ϕ : (X×S,r) →
B(X,S) to be the canonical map, where r = q × p is the product convergence
structure. Equip B(X,S) with the convergence quotient structure σ. Then



176 H. Boustique, P. Mikusiński and G. Richardson

K
σ
→ 〈(y,h)〉 iff there exist (x,g) ∼ (y,h) and H

r
→ (x,g) such that ϕ→H = K.

The space (B(X,S),σ) is investigated in section 4.

Remark 3.2. Fix a set X. the set of all convergence structures on X with the
ordering p ≤ q defined in section 2 is a complete lattice. Indeed, if (X,qj ) ∈

|CONV|, j ∈ J, then sup
j∈J

qj = q
1 is given by F

q
1

→ x iff F
qj
→ x, for each

j ∈ J. Dually, inf
j∈J

qj = q
0 is defined by F

q
0

→ x iff F
qj
→ x, for some j ∈ J.

It is easily verified that if ((X,qj ), (S,.,p),λ) ∈ AC for each j ∈ J, then both
((X,q1), (S,.,p),λ) and ((X,q0), (S,.,p),λ) belong to AC.

Theorem 3.3. Assume that ((X,q), (S,.,p),λ) ∈ AC. Then

(a) there exists a finest convergence structure qF on X such that
((X,qF ), (S,.,p),λ) ∈ AC

(b) there exists a coarsest convergence structure pc on S for which
((X,q), (S,.,pc),λ) ∈ AC

(c) ((B(X,S),σ), (S,.,p)) ∈ AC provided (S,.) is commutative and λ(.,g)
is an injection, for each g ∈ S.

Proof. (a): Define qF as follows: F
q

F

→ x iff there exist z ∈ X, G
p
→ g such

that x = λ(z,g) and F ≥ λ→(ż × G). Then (X,qF ) ∈ |CONV|. Indeed, ẋ
q

F

→ x
since x = λ(x,e) and ẋ = λ→(ẋ × ė). Hence (CS1) is satisfied. Clearly (CS2)
is valid, and (X,qF ) ∈ |CONV|.

It is shown that λ : (X,qF ) × (S,p) → (X,qF ) is continuous. Suppose that

F
q

F

→ x and H
p
→ h; then there exist z ∈ X, G

p
→ g such that x = λ(z,g) and

F ≥ λ→(ż ×G). Hence, F ×H ≥ λ→(ż ×G) ×H, and employing (a2), λ→(F ×
H) ≥ λ→(λ→(ż × G) × H) = [{λ({z} ×G.H) : G ∈ G,H ∈ H}] = λ→(ż × G.H).

Since G.H
p
→ g.h and λ(z,g.h) = λ(λ(z,g),h) = λ(x,h), it follows from the

definition of qF that λ→(F × H)
q

F

→ λ(x,h). Hence ((X,qF ), (S,.,p),λ) ∈ AC.
Assume that ((X,s), (S,.,p),λ) ∈ AC. It is shown that s ≤ qF . Suppose that

F
q

F

→ x; then there exist z ∈ X, G
p
→ g such that x = λ(z,g) and F ≥ λ→(ż×G).

Since λ→(ż × G)
s
→ λ(z,g), it follows that F

s
→ x and thus s ≤ qF . Hence

qF is the finest convergence structure on X such that ((X,qF ), (S,.,p),λ) ∈ AC.

(b): Define pc as follows: G
p

c

→ g iff for each F
q
→ x, λ→(F × G)

q
→ λ(x,g).

Then (S,pc) ∈ |CONV|. First, it is shown that (S,.,pc) ∈ |CSG|; that is,

if G
p

c

→ g and H
p

c

→ h, then G.H
p

c

→ g.h. Assume that F
q
→ x; then us-

ing (a2), λ→(F × G.H) = [{λ(F × G.H) : F ∈ F,G ∈ G,H ∈ H}] =
[{λ(λ(F × G) × H) : F ∈ F,G ∈ G,H ∈ H}] = λ→(λ→(F × G) × H).

It follows from the definition of pc that λ→(F × G)
q
→ λ(x,g), and thus

λ→(λ→(F × G) × H)
q
→ λ(λ(x,g),h) = λ(x,g.h). Hence G.H

p
c

→ g.h, and



Convergence semigroup actions: generalized quotients 177

thus (S,.,pc) ∈ |CSG|. According to the construction, pc is the coarsest con-
vergence structure on S such that λ : (X,q) × (S,pc) → (X,q) is continuous.

(c): Define λB : (B(X,S),σ) × (S,.,p) → (B(X,S),σ) by λB (〈(x,g)〉,h) =
〈(x,g.h)〉. It is shown that λB is a continuous action. Indeed, λB(〈(x,g)〉,e) =
〈(x,g)〉, and λB (λB (〈(x,g)〉,h),k) = λB (〈(x,g.h)〉,k) = 〈(x,g.h.k)〉 =
λB(〈(x,g)〉,h.k). Hence λB is an action. It remains to show that λB is

continuous. Suppose that K
σ
→ 〈(x,g)〉 and L

p
→ l. Since ϕ is a quotient

map in CONV, there exists H
r
→ (x1,g1) ∼ (x,g) such that ϕ

→H = K.
Then λ→B (K × L) = λ

→

B (ϕ
→H × L). Let K ∈ K and L ∈ L, and note

that λB (ϕ(H) × L) ⊆ λB(ϕ(π1(H) × π2(H)) × L) = ϕ(π1(H) × π2(H).L).

Hence λ→B (ϕ
→H × L) ≥ ϕ→(π→1 H × π

→

2 H.L)
σ
→ ϕ(x1,g1.l) = 〈(x1,g1.l)〉 =

λB(〈(x1,g1)〉, l) = λB (〈(x,g)〉, l). Therefore (B(X,S),S,λB ) ∈ AC. �

Remark 3.4. Let (X,q) ∈ |CONV| and let (C(X,X),c) denote the space
defined in section 2. Since c is the coarsest convergence structure for which
the evaluation map ω : (X,q) × (C(X,X),c) → (X,q) is continuous, this is
a particular case of Theorem 3.3(b), where λ = ω, (S,.,pc) = (C(X,X), .,c),
and the group operation is composition. Moreover, it is well-known that, in
general, there fails to exist a coarsest topology on C(X,X) for which ω : (X,q)×
C(X,X) → (X,q) is jointly continuous (even when q is a topology).

Assume that (X,S,λ) ∈ A; then λ is said to distinguish elements in S
whenever λ(x,g) = λ(x,h) for all x ∈ X implies that g = h. In this case,
define θ : S → C(X,X) by θ(g)(x) = λ(x,g), for each x ∈ X. Note that θ
is an injection iff λ separates elements in S. Moreover, θ is a homomorphism
whenever the operation in C(X,X) is k.l = l ◦ k is composition.

Theorem 3.5. Suppose that ((X,q), (S,.,p),λ) ∈ AC, and assume that λ dis-
tinguishes elements in S. Then the following are equivalent:
(a) θ : (S,p) → (C(X,X),c) is an embedding
(b) p = pc

(c) if G
p

6→ g, then there exists F
q
→ x such that λ→(F × G)

q

6→ λ(x,g).

Proof. (a) ⇒ (b): Assume that θ : (S,p) → (C(X,X),c) is an embedding.

According to Theorem 3.3(b), pc ≤ p. Suppose that G
p

c

→ g; then if F
q
→ x,

λ→(F × G)
q
→ λ(x,g). It is shown that θ→G

c
→ θ(g). Indeed, note that

ω→(F × θ→G) = [{ω(F × θ(G)) : F ∈ F,G ∈ G}] = [{λ(F × G) : F ∈ F,G ∈

G}] = λ→(F ×G)
q
→ λ(x,g) = ω(x,θ(g)). Hence θ→G

c
→ θ(g), and thus G

p
→ g.

Therefore p = pc.

(b) ⇒ (c): Verification follows directly from the definition of pc.

(c) ⇒ (a): Suppose that G
p
→ g and F

q
→ x. Since λ : (X,q) × (S,p) → (X,q)

is continuous, λ→(F × G)
q
→ λ(x,g). Hence ω→(F × θ→G) = λ→(F × G)

q
→



178 H. Boustique, P. Mikusiński and G. Richardson

λ(x,g) = ω(x,θ(g)), and thus θ→G
c
→ θ(g). Conversely, if G ∈ F(S) such that

θ→G
c
→ θ(g), then the hypothesis implies that G

p
→ g. Hence θ : (S,p) →

(C(X,X),c) is an embedding. �

Remark 3.6. The map θ given in Theorem 3.5 is called a continuous repre-
sentation of (S,.,p) on (X,q). Rath [10] discusses this concept in the context
of a group with (C(X,X), .,c) replaced by (H(X), .,γ), where (H(X), .) is the
group of all homeomorphisms on X with composition as the group operation,
and γ is the coarsest convergence structure making the operations of composi-
tion and inversion continuous.

Quite often it is desirable to consider modifications of convergence struc-
tures. For example, given (X,q) ∈ |CONV|, there exists a finest regular con-
vergence structure on X which is coarser than q [4]. The notation P q denotes
the P-modification of q. Generally, P represents a convergence space prop-
erty; however, it is convenient to include the case whenever Pq = q. Let
PCONV denote the full subcategory of CONV consisting of all the objects in
CONV that satisfy condition P . Condition P is said to be finitely produc-
tive(productive) provided that for each collection (Xj,qj ) ∈ |CONV|, j ∈ J,
P( ×

j∈J
qj ) = ×

j∈J
Pqj whenever J is a finite (arbitrary) set, respectively.

Theorem 3.7. Assume that FP : CONV → PCONV is a functor obey-
ing FP (X,q) = (X,Pq), FP (f) = f, and suppose that P is finitely produc-
tive. If ((X,q), (S,.,p),λ) ∈ AC and h : (T,.,ξ) → (S,.,p) is a contin-
uous homomorphism in CSG, then ((X,Pq), (T,.,Pξ)) ∈ AC; in particular,
((X,Pq), (S,.,Pp),λ) ∈ AC.

Proof. Given that ((X,q), (S,.,p),λ) ∈ AC, define Λ : (X,q) × (T,ξ) → (X,q)
by Λ(x,t) = λ(x,h(t)). Clearly Λ is an action; moreover, Λ is continuous.

Indeed, suppose that F
q
→ x and G

ξ
→ t; then Λ→(F × G) = [{Λ(F × G) : F ∈

F,G ∈ G}] = [{λ(F × h(G)) : F ∈ F,G ∈ G}] = λ→(F × h→G)
q
→ λ(x,h(t)) =

Λ(x,t). Therefore Λ is continuous.
Since FP is a functor and P is finitely productive, continuity of the operation

γ : (T,.,ξ) × (T,.,ξ) → (T,.,ξ), defined by γ(t1, t2) = t1.t2, implies continuity
of γ : (T,.,Pξ)×(T,.,Pξ) → (T,.,Pξ). Hence (T,.,Pξ) ∈ |CSG|. Likewise, Λ :
(X,Pq) × (T,Pξ) → (X,Pq) is continuous, and thus ((X,Pq), (T,.,Pξ), Λ) ∈
AC. �

Let (Sj, .,pj ) ∈ |CSG|, j ∈ J, and denote the product by (S,.,p) = ×
j∈J

(Sj, .,pj ).

The direct sum of (Sj, .), j ∈ J, is the subsemigroup of (S,.) defined by
⊕j∈JSj = {(gj) ∈ S : gj = ej for all but finitely many j ∈ J}. Denote
θj : Sj → ⊕j∈JSj to be the map θj (g) = (gk), where gj = g and gk = ek when-

ever k 6= j, and let θ : ⊕j∈JSj → ×
j∈J

Sj be the inclusion map. Define H
η
→ (gj )

in ⊕j∈JSj iff H ≥ θ
→

k1
G1.θ

→

k2
G2...θ

→

kn
Gn, where Gj

pkj
→ gkj in (Skj , .,pkj ) and



Convergence semigroup actions: generalized quotients 179

n ≥ 1. Then (⊕j∈JSj, .,η) ∈ |CSG|, and θ : (⊕j∈JSj, .,η) → (S,.,p) is a con-
tinuous homomorphism.

Theorem 3.8. Suppose that FP : CONV → PCONV is a functor satis-
fying FP (X,q) = (X,Pq), FP (f) = f, and P is productive. Assume that
((Xj,qj ), (Sj, .,pj ),λj ) ∈ AC for each j ∈ J. Then
(a) ( ×

j∈J
(Xj,Pqj ), ×

j∈J
(Sj, .,Ppj )) ∈ AC

(b) ( ×
j∈J

(Xj,Pqj ), (⊕j∈JSj, .,Pη)) ∈ AC.

Proof. (a): Denote (X,q) = ×
j∈J

(Xj,qj ), (S,.,p) = ×
j∈J

(Sj, .,pj ), and define λ :

(X,q) × (S,p) → (X,q) by λ((xj ), (gj )) = (λj (xj,gj)). Clearly λ is an action.
Then, according to Theorem 3.7 and the assumption that P is productive, it
suffices to show that ((X,q), (S,p),λ) ∈ AC. The latter follows from a routine
argument, and thus ( ×

j∈J
(Xj,Pqj ), ×

j∈J
(Sj, .,Ppj ),λ) ∈ AC.

(b): Since θ : (⊕Sj, .,η) → (S,.,p) is a continuous homomorphism in CSG and
P is productive, it follows from Theorem 3.7 that ( ×

j∈J
(Xj,Pqj), (⊕Sj, .,Pη)) ∈

AC. �

Corollary 3.9. Assume that FP : CONV → PCONV is a functor satisfying
FP (X,q) = (X,Pq), FP (f) = f, and P is finitely productive. Suppose that
((Xj,qj ), (Sj, .,pj )) ∈ AC for each j ∈ J. Denote (X,q) = ×

j∈J
(Xj,qj ) and

(S,.,p) = ×
j∈J

(Sj, .,pj). Then

(a) ((X,Pq), (S,.,Pp)) ∈ AC
(b) ((X,Pq), (⊕j∈JSj, .,Pη)) ∈ AC.

Verification of Corollary 3.9 follows the proof of Theorem 3.8 with the excep-
tion that since P is only finitely productive, (X,Pq) and ×

j∈J
(Xj,Pqj), as well

as (S,.,Pp) and ×
j∈J

(Sj, .,Ppj ), may differ. Of course equality holds whenever

the index set is finite. Choices of P that are finitely productive, and pre-
serve continuity when taking P-modifications include: locally compact, locally
bounded, regular, and first-countable. The property of being Choquet is pro-
ductive, and continuity is preserved under taking Choquet modifications.

4. Generalized Quotients

Recall that if ((X,q), (S,.,p),λ) ∈ AC, (S,.) is commutative, λ(.,g) is an
injection, then by Lemma 3.1, (x,g) ∼ (y,h) iff λ(x,h) = λ(y,g) is an equiv-
alence relation. Denote R = {((x,g), (y,h)) : (x,g) ∼ (y,h)}, r = q × p, and
ϕ : (X × S,r) → ((X × S)/ ∼,σ) the convergence quotient map defined by
ϕ(x,g) = 〈(x,g)〉. Then (B(X, S), σ):= ((X × S)/ ∼,σ) is called the gen-
eralized quotient space. Convergence space properties of (B(X,S),σ) are



180 H. Boustique, P. Mikusiński and G. Richardson

investigated in this section.
For ease of exposition, ((X,q), (S,.,p),λ) ∈ GQ denotes that ((X,q), (S,.,p),λ) ∈

AC, (S,.) is commutative, and λ(.,g) is an injection, for each g ∈ S. The gen-
eralized quotient space (B(X,S),σ) exists whenever ((X,q), (S,.,p),λ) ∈ GQ.

Theorem 4.1. Assume that ((X,q), (S,.,p),λ) ∈ GQ. Then the following are
equivalent:

(a) (X,q) is Hausdorff
(b) R is closed in ((X × S) × (X × S),r × r)
(c) (B(X,S),σ) is Hausdorff.

Proof. (a) ⇒ (b): Let πij denote the projection map defined by : πij : (X ×
S) × (X × S) → X × S where πij (((x,g), (y,h))) = (x,g) when i,j = 1, 2 and

πij (((x,g), (y,h))) = (y,h) when i,j = 3, 4. Suppose that H
r×r
→ ((x,g), (y,h))

and R ∈ H. Let H ∈ H; then H∩R 6= ∅, and thus there exists ((x1,g1), (y1,h1)) ∈
H ∩ R. Hence λ(x1,h1) = λ(y1,g1), and consequently λ((π1 ◦ π12)(H) × (π2 ◦
π34)(H)) ∩ λ((π1 ◦ π34)(H) × (π2 ◦ π12)(H)) 6= ∅, for each H ∈ H. It follows
that K := λ→((π1 ◦π12)

→H×(π2 ◦π34)
→H)∨λ→((π1 ◦π34)

→H×(π2 ◦π12)
→H)

exists. However, (π1 ◦ π12)
→H

q
→ x, (π2 ◦ π34)

→H
p
→ h, (π1 ◦ π34)

→H
q
→ y,

(π2 ◦ π12)
→H

p
→ g, and thus K

q
→ λ(x,h),λ(y,g). Since (X,q) is Hausdorff,

λ(x,h) = λ(y,g) and thus (x,g) ∼ (y,h). Therefore, ((x,g), (y,h)) ∈ R, and
thus R is closed.

(b) ⇒ (c): Assume that K
σ
→ 〈(yi,hi)〉, i = 1, 2. Since ϕ : (X × S,r) →

(B(X,S),σ) is a quotient map in CONV, there exist (xi,gi) ∼ (yi,hi) and

Hi
r
→ (xi,gi) such that ϕ

→Hi = K, i = 1, 2. Then for each Hi ∈ Hi,
ϕ(H1) ∩ ϕ(H2) 6= ∅ and thus there exists (si, ti) ∈ Hi such that (s1, t1) ∼

(s2, t2), i = 1, 2. Hence the least upper bound filter L := (H1 × H2) ∨ Ṙ exists,

and L
r×r
→ ((x1,g1), (x2,g2)). Since R is closed, (x1,g1) ∼ (x2,g2) and thus

〈(y1,h1)〉 = 〈(y2,h2)〉. Therefore (B(X,S),σ) is Hausdorff.

(c) ⇒ (a): Suppose that (B(X,S),σ) is Hausdorff and F
q
→ x,y. Then

ϕ→(F ×ė)
σ
→ 〈(x,e)〉,〈(y,e)〉, and thus (x,e) ∼ (y,e). Therefore, x = λ(x,e) =

λ(y,e) = y, and thus (X,q) is Hausdorff. �

Conditions for which (B(X,S),σ) is T1 are given below. In the topological
setting, sufficient conditions in order for the generalized quotient space to be
T2 are given in [1] whenever (S,.) is equipped with the discrete topology.

Theorem 4.2. Suppose that ((X,q), (S,.,p),λ) ∈ GQ. Then (B(X,S),σ) is
T1 iff ϕ

−1(〈(y,h)〉) is closed in (X × S,r), for each (y,h) ∈ X × S.

Proof. The ”only if” is clear since {〈(y,h)〉} is closed and ϕ is continuous.
Conversely, assume that ϕ−1(〈(y,h)〉) is closed, for each (y,h) ∈ X × S, and



Convergence semigroup actions: generalized quotients 181

suppose that ˙〈(x,g)〉
σ
→ 〈(y,h)〉. Since ϕ is a quotient map in CONV, there exist

(s,t) ∼ (y,h) and H
r
→ (s,t) such that ϕ→H = ˙〈(x,g)〉. Then ϕ−1(〈(x,g)〉) ∈

H, and thus (s,t) ∈ clrϕ
−1(〈(x,g)〉) = ϕ−1(〈(x,g)〉). Hence (x,g) ∼ (s,t) ∼

(y,h), and thus 〈(x,g)〉 = 〈(y,h)〉. Therefore (B(X,S),σ) is T1. �

Corollary 4.3. Assume that ((X,q), (S,.,p),λ) ∈ GQ, and let p denote the
discrete topology. Then (B(X,S),σ) is T1 iff (X,q) is T1.

Proof. Suppose that (B(X,S),σ) is T1 and ẋ
q
→ y. Then ˙(x,e)

r
→ (y,e), and

thus ˙〈(x,e)〉 = ϕ→( ˙(x,e))
σ
→ 〈(y,e)〉. It follows that 〈(x,e)〉 = 〈(y,e)〉 and

hence x = y. Therefore (X,q) is T1.

Conversely, assume that (X,q) is T1 and (y,h) ∈ clrϕ
−1(〈(x,g)〉). Then

there exists H
r
→ (y,h) such that ϕ−1(〈(x,g)〉) ∈ H, π→1 H

q
→ y, π→2 H

p
→ h,

and since p is the discrete topology, choose H ∈ H for which π2(H) = {h}
and ϕ(H) = {〈(x,g)〉}. If (s,t) ∈ H, then (s,t) ∼ (x,g), t = h, and thus

λ(s,g) = λ(x,h). Hence λ(π1(H) × {g}) = {λ(x,h)}, and thus
˙λ(x,h) =

λ→(π→1 H × ġ)
q
→ λ(y,g). Then λ(x,h) = λ(y,g), (x,g) ∼ (y,h), and thus

ϕ−1(〈(x,g)〉) is r-closed. Hence it follows from Theorem 4.2 that (B(X,S),σ)
is T1. �

Corollary 4.4 ([1]). Suppose that the hypotheses of Corollary 4.3 are satisfied
with the exception that (X,q) is a topological space and B(X,S) is equipped
with the quotient topology τ. Then (B(X,S),τ) is T1 iff (X,q) is T1.

Proof. It follows from Theorem 2 [2] that since ϕ : (X × S,r) → (B(X,S),σ)
is a quotient map in CONV, ϕ : (X × S,r) → (B(X,S), tσ) is a topological
quotient map, where tσ is the largest topology on X × S which is coarser than
σ. Moreover, τ = tσ , and A ⊆ B(X,S) is σ-closed iff it is τ-closed. Hence the
desired conclusion follows from Corollary 4.3. �

An illustration is given to show that the generalized quotient space may fail
to be T1 even though (X,q) is a T1 topological space.

Example 4.5. Denote X = (0, 1), q the cofinite topology on X, and define
f : X → X by f(x) = ax, where 0 < a < 1 is fixed. Let S = {fn : n ≥ 0},
where f0 = idX and f

n denotes the n-fold composition of f with itself. Then
(S,.) ∈ |SG| is commutative with composition as the operation. Also equip
(S,.) with the cofinite topology p. It is shown that the operation γ : (S,p) ×
(S,p) → (S,p) defined by γ(g,h) = g.h := h ◦ g is continuous at (fm,fn).
Define C = {fk : k ≥ k0}; then {f

m+n} ∪ C is a basic p-neighborhood of
fm+n, where k0 ≥ 0. Observe that if A = {f

m} ∪ C and B = {fn} ∪ C, then
γ(A × B) ⊆ C ∪ {fm+n}. Therefore γ is continuous, and (S,.,p) ∈ |CSG|.

Define λ : X × S → X by λ(x,g) = g(x), for each x ∈ X, g ∈ S, and note
that λ is an action. It is shown that λ : (X,q) × (S,p) → (X,q) is continuous
at (x0,f

n) in X × S. A basic q-neighborhood of λ(x0,f
n) = fn(x0) is of the



182 H. Boustique, P. Mikusiński and G. Richardson

form W = X − F , where fn(x0) /∈ F and F is a finite subset of X. Let y0
be the smallest member of F , and choose k0 to be a natural number such that
ak0 < y0. Then for each k ≥ k0, f

k(x) = akx < y0 for each x ∈ X. Since f
n

is injective, F0 = (f
n)−1(F) is a finite subset of X. Then U = X − F0 is a

q-neighborhood of x0, V = {f
n}∪{fk : k ≥ k0} is a p-neighborhood of f

n, and
λ(U × V ) ⊆ W . Indeed, if x ∈ U and k ≥ k0, then λ(x,f

k) = fk(x) < y0, and
thus fk(x) ∈ W . Further, if x ∈ U, then fn(x) /∈ F , and hence fn(x) ∈ W . It
follows that λ(U × V ) ⊆ W , and thus λ is a continuous action.

It is shown that ϕ−1(〈(x0, idX )〉) is not closed in (X × S,r). Note that
(x,fn) ∈ ϕ−1(〈(x0, idX )〉) iff idX (x) = f

n(x0). Hence ϕ
−1(〈(x0, idX )〉) =

{(fn(x0),f
n) : n ≥ 0}. Since idX = f

0 > f1 > f2 > ..., it easily follows
that clrϕ

−1(〈(x0, idX )〉) = X ×S, and thus ϕ
−1(〈(x0, idX )〉) is not r-closed. It

follows from Theorem 4.2 that (B(X,S),σ) is not T1 even though both (X,q)
and (S,p) are T1 topological spaces.

A continuous surjection f : (X,q) → (Y,p) in CONV is said to be proper

map provided that for each ultrafilter F on X, f→F
p
→ y implies that F

q
→ x,

for some x ∈ f−1(y). Proper maps in CONV are discussed in [3]; in particular,
proper maps preserve closures. A proper convergence quotient map is called a
perfect map [4].

Remark 4.6. Assume that ((X,q), (S,.,p),λ) ∈ GQ, (X,q) and (S,p) are
regular, and ϕ : (X×S,r) → ((B(X,S),σ) is a perfect map. Then (B(X,S),σ)

is also regular. Indeed, suppose that H ∈ F(B(X,S)) such that H
σ
→ 〈(y,h)〉.

Since ϕ is a quotient map in CONV, there exists (x,g) ∼ (y,h) and K
r
→

(x,g) such that ϕ→K = H. Moreover, the regularity of (X × S,r) implies

that clrK
r
→ (x,g). Since ϕ is a proper map and thus preserves closures,

ϕ→(clrK) = clσϕ
→K = clσH

σ
→ 〈(y,h)〉. Hence (B(X,S),σ) is regular.

The proof of the following result is straightforward to verify.

Lemma 4.7. Suppose that (S,.,p) ∈ |CSG| and (T,.) ∈ |SG|. Assume that
f : (S,.,p) → (T,.,σ) is both a homomorphism and a quotient map in CONV.
Then (T,.,σ) ∈ |CSG|.

Assume that ((X,q), (S,.,p),λ) ∈ AC. Recall that λ distinguishes elements
in S whenever λ(x,g) = λ(x,h) for each x ∈ X implies g = h. This property
was needed in the verification of Theorem 3.5. In the event that λ fails to
distinguish elements in S, define g ∼ h iff λ(x,g) = λ(x,h) for each x ∈ X.
Then ∼ is an equivalence relation on S; denote S1 = S/ ∼= {[g] : g ∈ S}, and
define the operation [g].[h] = [g.h], for each g,h ∈ S. The operation is well
defined and (S1, .) ∈ |SG|. Let p1 denote the quotient convergence structure
on S1 determined by ρ : (S,p) → S1, where ρ(g) = [g]. Then ρ : (S,.) → (S1, .)
is a homomorphism, and it follows from Lemma 4.7 that (S1, .,p1) ∈ |CSG|.
Define λ1 : X × S1 → X by λ1(x, [g]) = λ(x,g).



Convergence semigroup actions: generalized quotients 183

Theorem 4.8. Assume ((X,q), (S,.,p),λ ∈) GQ, λ fails to distinguish el-
ements in S, and let (B(X × S),σ), (B(X × S1),σ1) denote the generalized
quotient spaces corresponding to (X × S,r) and (X × S1,r1), where r = q × p
and r1 = q × p1. Then

(a) λ1 : (X × S1,r1) → (X,q) is a continuous action
(b) λ1 separates elements in S1
(c) (B(X,S),σ) and (B(X,S1),σ1) are homeomorphic.

Proof. (a): It is routine to verify that λ1 is an action. Let us show that λ1
is continuous. Suppose that F

q
→ x and G

p1
→ [g]; then since p1 is a quotient

structure in CONV, there exists G1
p
→ g1 ∼ g such that ρ

→G1 = G. Hence
λ→1 (F × G) = λ

→

1 (F × ρ
→G1) = [{λ1(F × ρ(G1)) : F ∈ F,G1 ∈ G1}] =

[{λ(F × G1) : F ∈ F,G1 ∈ G1}] = λ
→(F × G1)

q
→ λ(x,g1) = λ1(x, [g]), and

thus λ1 is continuous.

(b): Suppose that λ1(x, [g]) = λ1(x, [h]) for each x ∈ X. Then λ(x,g) = λ(x,h)
for each x ∈ X, and thus [g] = [h]. Hence λ1 distinguishes elements in S1.

(c): It easily follows that the diagram below is commutative:

X × S
ϕ1
- B(X,S)

X × S1

ψ1

? ϕ2
- B(X,S1)

ψ2

?

where ϕ1, ϕ2 are quotient maps, ψ1(x,g) = (x, [g]), and ψ2(〈x,g〉) = 〈(x, [g])〉.
Moreover, ψ2 is an injection. Indeed, assume that 〈(x, [g])〉 = ψ2(〈(x,g)〉) =
ψ2(〈(y,h)〉) = 〈(y, [h])〉; then λ1(x, [h]) = λ1(y, [g]) and thus λ(x,h) = λ(y,g).
Hence 〈(x,g)〉 = 〈(y,h)〉 and ψ2 is an injection. Clearly ψ2 is a surjection.

It is shown that ψ2 is continuous. Indeed, suppose that H
σ
→ 〈(y,h)〉; then

there exist (x,g) ∼ (y,h) and K
r
→ (x,g) such that ϕ→1 K = H. Since the

diagram above commutes with ψ1 and ϕ2 continuous, it follows that ψ
→

2 H =

(ψ2 ◦ϕ1)
→K = (ϕ2 ◦ψ1)

→K
σ1
→ (ϕ2 ◦ψ1)(x,g) = (ψ2 ◦ϕ1)(x,g) = ψ2(〈(x,g)〉) =

ψ2(〈(y,h)〉). Hence ψ2 is continuous.

Finally, let us show that ψ−12 is continuous. Assume that H
σ1
→ 〈(y, [h])〉.

Since ϕ2 is a quotient map, there exist (x, [g]) ∼ (y, [h]) and K
r1
→ (x, [g]) such

that ϕ→2 K = H. In particular, F = π
→

1 K
q
→ x and G = π→2 K

p1
→ [g]. Since

ρ : (S,p) → (S1,p1) is a quotient map, there exist g1 ∼ g and G1
p
→ g1 such that

ρ→G1 = G. Then F×G1
r
→ (x,g1), and thus ψ

→

1 (F×G1) = F×ρ
→G1 = F×G ≤

K. Hence (ϕ2 ◦ ψ1)
→(F × G1) ≤ ϕ

→

2 K = H, and since the diagram commutes,

ψ←2 H ≥ (ψ
−1
2 ◦ ϕ2 ◦ ψ1)

→(F × G1) = ϕ
→

1 (F × G1)
σ
→ 〈(x,g)〉 = ψ−12 (〈(y, [h])〉).

Therefore ψ2 is a homeomorphism. �



184 H. Boustique, P. Mikusiński and G. Richardson

Sufficient conditions in order for (X,q) to be embedded in (B(X,S),σ) are
presented below.

Theorem 4.9. Suppose that ((X,q), (S,.,p),λ) ∈ GQ. Define β : (X,q) →
(B(X,S),σ) by β(x) = 〈(x,e)〉, for each x ∈ X. Then

(a) β is a continuous injection
(b) β is an embedding provided that (X,q) is a Choquet space, p is discrete,

and λ is a proper map.

Proof. (a): Clearly β is an injection. Next, assume that F
q
→ x; then β→F =

[{β(F) : F ∈ F}] = [{ϕ(F × {e}) : F ∈ F}] = ϕ→(F × ė)
σ
→ ϕ(x,e) = β(x).

Therefore β is continuous.

(b): First, suppose that F is an ultrafilter on X such that β→F
σ
→ β(x) =

〈(x,e)〉. Since ϕ : (X × S,r) → (B(X,S),σ) is a quotient map in CONV,

there exist (y,g) ∼ (x,e) and K
r
→ (y,g) such that ϕ→K = β→F. Denote

F1 = π
→

1 K
q
→ y and G1 = π

→

2 K
p
→ g. Since p is the discrete topology, G1 = ġ,

and thus K ≥ π→1 K×π
→

2 K = F1 × ġ. Let F1 ∈ F1; then ϕ
→(F1 × ġ) ≤ ϕ

→K =
β→F implies that there exists F ∈ F such that β(F) ⊆ ϕ(F1 × {g}). If z ∈ F ,
then β(z) = 〈(z,e)〉 = 〈(z1,g)〉, for some z1 ∈ F1, and thus λ(z,g) = λ(z1,e) =

z1 ∈ F1. It follows that λ(F × {g}) ⊆ F1, and thus λ
→(F × ġ) ≥ F1

q
→ y.

Since F × ġ is an ultrafilter on X × S and λ is a proper map, F × ġ
r
→ (s,t),

for some (s,t) ∈ λ−1(y). Then F
q
→ s and g = t since p is discrete. It follows

that λ(y,e) = y = λ(s,t) = λ(s,g), and thus (s,e) ∼ (y,g). As shown above,

(y,g) ∼ (x,e), and thus (x,e) ∼ (s,e). Therefore x = s, and F
q
→ x.

Finally, let F be any filter on X such that β→F
σ
→ β(x). If H is any

ultrafilter on X containing F, then β→H
σ
→ β(x), and from the previous

case, H
q
→ x. Since (X,q) is a Choquet space, F

q
→ x and hence β is an

embedding. �

Assume that ((X,q), (S,.,p),λ) ∈ GQ, (X,q̄) is the finest Choquet space
such that q̄ ≤ q, r̄ = q̄ ×p, and let σ̄ denote the quotient convergence structure
on B(X,S) determined by ϕ : (X × S, r̄) → B(X,S).

Corollary 4.10. Assume ((X,q), (S,.,p),λ) ∈ GQ, p is discrete, and λ is a
proper map. Then, using the above notations, β : (X,q̄) → (B(X,S), σ̄) is an
embedding.

Proof. It follows from Theorem 3.7 that ((X,q̄), (S,.,p),λ) ∈ AC. Since q and
q̄ agree on ultrafilter convergence, λ : (X,q̄) × (S,p) → (X,q̄) is also a proper
map, and (X,q̄) is a Choquet space. Then according to Theorem 4.9, β :
(X,q̄) → (B(X × S), σ̄) is an embedding. �



Convergence semigroup actions: generalized quotients 185

Let us conclude by showing that the generalized quotient of a product
is homeomorphic to the product of the generalized quotients. Assume that
((Xj,qj ), (Sj, .,pj ),λj ) ∈ GQ, for each j ∈ J. Let (X,q) = ×

j∈J
(Xj,qj ) and

(S,.,p) = ×
j∈J

(Sj, .,pj) denote the product spaces, and define λ : X × S → X

by λ((xj ), (gj )) = (λj (xj,gj )). According to Corollary 3.9, ((X,q), (S,.,p),λ) ∈
AC. Moreover, since each (Sj, .,pj ) is commutative and λj (.,g) is an injection
for each j ∈ J, (S,.,p) is commutative and λ(.,g) is an injection. Hence
((X,q), (S,.,p),λ) ∈ GQ. Let ϕj : (Xj,qj) × (Sj, .,pj) → (B(Xj,Sj),σj )
denote the convergence quotient map, rj = qj × pj, ϕ = ×

j∈J
ϕj , for each

j ∈ J. Since the product of quotient maps in CONV is again a quotient
map, ϕ : ×

j∈J
(Xi × Sj,rj ) → ×

j∈J
(B(Xj,Sj ),σj ) is also a quotient map. Denote

σ = ×
j∈J

σj .

Define ((xj ), (gj )) ∼ ((yj ), (hj )) in X × S iff λ((xj ), (hj )) = λ((yj ), (gj )).
This is an equivalence relation on X × S, and it follows from the definition
of λ that ((xj ), (gj )) ∼ ((yj ), (hj )) iff (xj,gj ) ∼ (yj,hj ), for each j ∈ J.
Let (B(X,S), Σ) denote the corresponding generalized quotient space, where
Φ : (X × S,r) → (B(X,S), Σ) is the quotient map and r = ×

j∈J
rj .

Theorem 4.11. Suppose that ((Xj,qj ), (Sj, .,pj),λj ) ∈ GQ, for each j ∈ J.
Then, employing the notations defined above, ×

j∈J
(B(Xj,Sj ),σj ) and (B(X,S), Σ)

are homeomorphic.

Proof. Consider the following diagram:

×
j∈J

(Xj × Sj,rj )
δ
- (X × S,r)

×
j∈J

(B(Xj,Sj ),σj )

ϕ

? ∆
- (B(X,S), Σ),

Φ

?

where δ(((xj,gj)j )) = ((xj ), (gj )) and ∆((〈(xj,gj )〉j )) = 〈((xj ), (gj ))〉. Then δ
is a homeomorphism, and the diagram commutes. Note that ∆ is a bijection.
Indeed, if ∆((〈(xj,gj )〉j )) = ∆((〈(yj,hj )〉j )), then ((xj ), (gj )) ∼ ((yj ), (hj ))
and thus (xj,gj) ∼ (yj,hj), for each j ∈ J. Hence 〈(xj,gj)〉j = 〈yj,gj〉j for
each j ∈ J, and thus ∆ is an injection. Clearly ∆ is a surjection.

It is shown that ∆ is continuous. Assume that H
σ
→ (〈(yj,hj )〉j ); then since

ϕ is a quotient map, there exist ((xj ), (gj )) ∼ ((yj ), (hj )) and K
r
→ ((xj,gj )j )

such that ϕ→K = H. However, the diagram commutes, and thus ∆→H =

(∆◦ϕ)→K = (Φ◦δ)→K
Σ
→ Φ((xj ), (gj )) = Φ((yj ), (hj )) = 〈((yj ), (hj ))〉. Hence

∆ is continuous.
Conversely, suppose that H

Σ
→ 〈((yj ), (hj ))〉; then since Φ is a quotient map,



186 H. Boustique, P. Mikusiński and G. Richardson

there exist ((xj ), (gj )) ∼ ((yj ), (hj )) and K
r
→ ((xj ), (gj )) such that Φ

→K = H.
Using the fact that δ is a homeomorphism and that the diagram commutes,

∆←H = (ϕ ◦ δ−1)→K
σ
→ ϕ((xj,gj )j ) = ϕ((yj,hj )j ) = (〈(yj,hj )〉j ), and thus

∆−1 is continuous. Therefore ∆ is a homeomorphism. �

Remark 4.12. In general, quotient maps are not productive in the category
of all topological spaces with the continuous maps as morphisms. Whether or
not Theorem 4.11 is valid in the topological context is unknown to the authors.

References

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[3] D. Kent and G. Richardson, Open and proper maps between convergence spaces, Czech.

Math. J. 23(1973), 15–23.
[4] D. Kent and G. Richardson, The regularity series of a convergence space, Bull. Austral.

Math. Soc. 13 (1975), 21–44.
[5] M. Khosravi, Pseudoquotients: Construction, applications, and their Fourier transform,

Ph.D. dissertation, Univ. of Central Florida, Orlando, FL, 2008.
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Received November 2008

Accepted November 2009

Gary Richardson (garyr@mail.ucf.edu)
Department of Mathematics, University of Central Florida,Orlando, FL 32816,
USA, fax: (407) 823-6253, tel: (407) 823-2753