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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 9, No. 2, 2008

pp. 205-212

On the topology of generalized quotients

Józef Burzyk, Cezary Ferens and Piotr Mikusiński

Abstract. Generalized quotients are defined as equivalence classes
of pairs (x, f ), where x is an element of a nonempty set X and f is an
element of a commutative semigroup G acting on X. Topologies on X
and G induce a natural topology on B(X, G), the space of generalized
quotients. Separation properties of this topology are investigated.

2000 AMS Classification: Primary 54B15, Secondary 20M30, 54D55.

Keywords: Generalized quotients, semigroup acting on a set, quotient topol-
ogy, Hausdorff topology.

1. Preliminaries

Let X be a nonempty set and let S be a commutative semigroup acting on
X injectively. For (x,ϕ), (y,ψ) ∈ X × S we write

(x,ϕ) ∼ (y,ψ) if ψx = ϕy.

This is an equivalence relation in X × S. Finally, we define

B(X,S) = (X × S)/∼,

the set of generalized quotients. The equivalence class of (x,ϕ) will be denoted
by x

ϕ
.

Elements of X can be identified with elements of B(X,S) via the embedding
ι : X → B(X,S) defined by ι(x) = ϕx

ϕ
, where ϕ is an arbitrary element of S.

The action of G can be extended to B(X,S) via ϕx
ψ

= ϕx
ψ

. If ϕx
ψ

= ι(y), for

some y ∈ X, we will write ϕx
ψ
∈ X and ϕ x

ψ
= y. For instance, we have ϕx

ϕ
= x.

Other properties of generalized quotients and several examples can be found
in [2] and [4].

If X is a topological space and G is a commutative semigroup of continuous
maps acting on X, equipped with its own topology, then we can define the
product topology on X × G and then the quotient topology on B(X,S) =
(X × G)/∼.



206 J. Burzyk, C. Ferens and P. Mikusiński

It is easy to show that the embedding ι : X → B(X,S) is continuous.
Moreover, the map x

ψ
7→ ϕx

ψ
is continuous for every ϕ ∈ G. These and other

topological properties of generalized quotients can be found in [1].
In this note we will always assume that the topology on G is discrete. In

most examples, it is a natural assumption.
Let Y be a topological space and let ∼ be an equivalence relation. If y ∈ Y ,

then by [y] we denote the equivalence class of y, that is, [y] = {w ∈ Y : w ∼ y}.
The map q : Y → Y/∼, defined by q(y) = [y], is called the quotient map. A
subset U ⊂ Y is called saturated if y ∈ U implies [y] ⊂ U. In other words, U
is saturated if U = q−1(q(U)). Let Z = Y/∼. A set V ⊂ Z is open (in the
quotient topology) if and only if V = q(U) for some open saturated U ⊂ Y .

Whenever convenient, we use convergence arguments. The sequential con-
vergence defined by the topology of B(X,G) is not easily characterized. The
following theorem is often useful.

Theorem 1.1. Let xn
ϕn

∈ B(X,G), n ∈ N. If there exist a ψ ∈ G and a y ∈ X

such that xn
ϕn

= yn
ψ

, for all n ∈ N, and yn → y in the topology of X, then
xn
ϕn

→ y
ψ

in the topology of B(X,G).

Proof. If U is an open neighborhood of y
ψ

in B(X,G), then (y,ψ) ∈ q−1(U).

Since q−1(U) is open in X × G, there exists an open V ⊂ X such that (y,ψ) ∈
V ×{ψ} ⊂ q−1(U). But then yn ∈ V for almost all n ∈ N, because yn → y in the
topology of X. Hence, (yn,ψ) ∈ q

−1(U) for almost all n ∈ N or, equivalently
yn
ψ

= xn
ϕn

∈ U for almost all n ∈ N. �

In this note we investigate some separation properties of the topology of
B(X,G).

2. General separation properties

We are interested in the general question whether a separation property of
X is inherited by B(X,G). First we consider T1.

Theorem 2.1. If X is T1 and the topology of G is discrete, then B(X,G) is
T1.

Proof. If x
ϕ
∈ B(X,G), then (X × G) \ q−1

(

x
ϕ

)

is an open saturated subset of

X × G. �

Now we give an example of a Banach space X and a semigroup G of contin-
uous injections on X for which B(X,G) is not Hausdorff.

If f,g : R → R and the set {t ∈ R : f(t) 6= g(t)} is meager in the usual
topology of R, then we will write f ≃ g. Let B(R) be the space of all bounded
real-valued functions on R and let X = B(R)/≃. With respect to the norm

‖[f]‖ = inf{‖g‖∞ : g ≃ f}

X is a Banach space. Let

G = {[f] ∈ X : {t ∈ R : f(t) = 0} is a meager set in R} .



On the topology of generalized quotients 207

Then G is a semigroup of injections acting on X by pointwise multiplication.
Note that B(X,G) can be identified with RR/≃. To show that the topology of
B(X,G) is not Hausdorff we need two simple lemmas. In what follows, we will
not distinguish between functions and equivalence classes of functions. The
indicator function of a set A will be denoted by IA.

Lemma 2.2. If (An) is a sequence of subsets of R such that An ⊂ An+1, for
each n ∈ N, and R \

⋃∞
n=1 An is meager, then for each f ∈ C(R) the sequence

fn = fIAn is convergent to f in B(X,G).

Proof. Define a function g : R → R as follows

g(t) =

{

1 if t ∈ A1,
1
n

if t ∈ An \ An−1.

It is easy to see that fng → fg in X. Consequently fn → f in B(X,G). �

Corollary 2.3. If a set U ⊂ B(X,G) is sequentially open and f
g
∈ U, then for

each r ∈ R there exists a open neighborhood V ⊂ R of r such that
fIR\V
g

∈ U.

Lemma 2.4. If (An) is a sequence of subsets of R such that An+1 ⊂ An, for
each n ∈ N, and the set

⋂∞
n=1 An is meager, then for each f ∈ X the sequence

(fn), where fn = fIAn , is convergent to 0 in B(X,G).

Proof. Use

g(t) =

{

1 if t /∈ A1,
1
n

if t ∈ An \ An+1.

�

Theorem 2.5. If U is a nonempty sequentially open subset of B(X,G), then
U is sequentially dense in B(X,G).

Proof. It is enough to prove that there exists a sequence Fn ∈ U such that
Fn → 0 in B(X,G). Consider an arbitrary element f/g ∈ U and assume that
(rn) is a sequence of all rational numbers. Then, by Corollary 2.3, there exits
a neighborhood V1 of r1 such, that

F1 =
fIR\V1
g

∈ U.

Next we find a neighborhood V2 of r2 such, that

F2 =
fIR\(V1∪V2)

g
∈ U.

By induction, we construct a sequence Vn ⊂ R such that Vn is a neighborhood
of rn and

Fn =
fIR\(V1∪...∪Vn)

g
∈ U.

The set
⋃∞
n=1 Vn is open and dense in R. Hence, the complement of

⋃∞
n=1 Vn is

a meager set. By Lemma 2.4, fIR\(V1∪...∪Vn) → 0 in B(X,G), and consequently
Fn → 0 in B(X,G). �



208 J. Burzyk, C. Ferens and P. Mikusiński

Since X in this example is a Banach space, no separation property of X above
T1 will be inherited by the topology of B(X,G) without additional assumptions.
In the remaining part of this note we give examples of theorems that discribe
special situations in which the topology of B(X,G) is Hausdorff.

3. Hausdorff property in special cases

First we introduce some notation and make some useful observations. If
U ⊂ X × G, then U =

⋃

ϕ∈G Uϕ × {ϕ}, where Uϕ ⊂ X. For every ψ ∈ G let
Πψ : X × G → X be the projection defined by

Πψ





⋃

ϕ∈G

Uϕ × {ϕ}



 = Uψ.

If A ⊂ X × G, then the smallest saturated set containing A will be denoted by
ΣA. We have the following straightforward characterization on ΣA.

Proposition 3.1. If A ⊂ X × G, then

ΣA =
⋃

ϕ,ψ∈G

ϕ−1ψΠϕA × {ψ}.

In other words, for every ψ ∈ G, we have

ΠψΣA =
⋃

ϕ∈G

ϕ−1ψΠϕA.

Corollary 3.2. A set A ⊂ X × G is saturated if and only if

ϕ−1ψΠϕA ⊂ ΠψA

for every ϕ,ψ ∈ G.

Theorem 3.3. If X is Hausdorff and every ϕ ∈ G is an open map, then
B(X,G) is Hausdorff.

Proof. Let x
ϕ

and y
ψ

be two distinct elements of B(X,G). It suffices to find

open and saturated subsets of X × G that separate (x,ϕ) and (y,ψ). Since
ψx 6= ϕy and X is Hausdorff, there exist open and disjoint sets U,V ⊂ X such
that ψx ∈ U and ϕy ∈ V . Define

A = ψ−1U × {ϕ} and B = ϕ−1V × {ψ}.

Consider the sets ΣA and ΣB. By Proposition 3.1, ΣA and ΣB are open sets.
If (z,γ) ∈ ΠγΣA, then z ∈ ϕ

−1γψ−1U, again by Proposition 3.1. This means
that ϕz = γψ−1u for some u ∈ U. Hence, (z,γ) ∼ (ψ−1u,ϕ). Similarly, if
(z,γ) ∈ ΠγΣB, there exists a v ∈ V such that (z,γ) ∼ (ϕ

−1v,ψ). Therefore,
(ψ−1u,ϕ) ∼ (ϕ−1v,ψ), which implies u = v, contradicting U ∩ V = ∅. �

For topological spaces X and Y , by C(X,Y ) we denote the space of continu-
ous maps from X to Y . For a continuous ϕ : X → X, by ϕ∗ : C(X,Y ) →
C(X,Y ) we denote the adjoint map, that is, (ϕ∗f)x = f(ϕx) where f ∈
C(X,Y ).



On the topology of generalized quotients 209

Theorem 3.4. Let X be a topological space, G a commutative semigroup of
continuous injections from X into X, equipped with the discrete topology, such
that ϕ(X) is dense in X for all ϕ ∈ G. Let Y be a Hausdorff space and let
F ⊂ C(X,Y ) be such that F separates points in X and for every ϕ ∈ G we
have F ⊂ ϕ∗(F). Then the topology of B(X,G) is Hausdorff.

Proof. First note that, since ϕ(X) is dense in X, ϕ∗ is a injection. For f ∈ F
and ϕ ∈ G define fϕ to be the unique function in F such that ϕ

∗fϕ = f. Then,
for any ϕ,ψ ∈ G, we have ψ∗fψ = f = (ϕψ)

∗fϕψ and hence ψ
∗fψ = ψ

∗ϕ∗fϕψ.
Since ψ∗ is injective, we have fψ = ϕ

∗fϕψ. Thus, fψ(x) = ϕ
∗fϕψ(x) = fϕψ(ϕx)

for any x ∈ X.
Consider two distinct elements F1 and F2 of B. Without loss of generality,

we can assume that F1 =
x1
ϕ

and F2 =
x2
ϕ

, for some x1 6= x2. There exists an

f ∈ F such that f(x1) 6= f(x2). Let Ω1, Ω2 ⊂ Y be open disjoint neighborhoods
of f(x1) and f(x2), respectively. For every ψ ∈ G let

Uψ = ϕ
−1

(

f−1
ψ

(Ω1)
)

and Vψ = ϕ
−1

(

f−1
ψ

(Ω2)
)

.

We will show that

U =
⋃

ψ∈G

Uψ × {ψ} and V =
⋃

ψ∈G

Vψ × {ψ}

are disjoint saturated open sets that separate (x1,ϕ) and (x2,ϕ). It suffices to
prove that the sets are saturated. Since the sets are defined the same way, we
will only prove it for U. Suppose x ∈ Uψ and (x,ψ) ∼ (y,γ). Then γx = ψy
and

fγ(ϕy) = fψγ(ϕψy) = fψγ(ϕγx) = fψ(ϕx) ∈ Ω1.

Thus y ∈ Uγ. �

Example 3.5. Let X = {x ∈ C(R) : x(0) = 0}, with the topology of uniform
convergence on compact sets, and let G = {Λn : n ∈ N0}, where Λx(t) =
∫ t

0
x(s) ds and N0 denotes the set of all nonnegative integers. To show that

the topology of B(X,G) is Hausdorff we use Theorem 3.4 with Y = R and
F = {f ∈ D(R) : f 6= 0}, where D(R) is the space of smooth functions with
compact support. If f ∈ F and x ∈ X, then we define f(x) =

∫ ∞

−∞
f(t)x(t) dt.

Clearly, Λn is injective and Λn(X) is dense in X for every n ∈ N. Moreover,
F separates points in X. If f ∈ F and n ∈ N, then there exists a g ∈ F such
that f(x) = g(Λnx) for every x ∈ X, namely g = (−1)nf(n). Thus all the
assumptions of the theorem are met.

The assumption that x(0) = 0, in the definition of X, may seem artificial.
It is made for convenience and it does not affect the final result. Note that for
any x ∈ C(R) we have x

Λn
= Λx

Λn+1
and Λx ∈ X. One can prove that, in general,

B(X,G) = B(gX,G) for any g ∈ G (see [1]).

In the next theorem we assume that G is generated by a single function,
that is, G = {ϕn : n ∈ N0}.



210 J. Burzyk, C. Ferens and P. Mikusiński

Proposition 3.6. Let G = {ϕn : n ∈ N0} and A ⊂ X × G. A is saturated if
and only if, for all i,j ∈ N0,

(3.1) z ∈ ΠiA if and only if ϕ
jz ∈ Πi+jA,

where Πk = Πϕk .

Proof. Assume that (3.1) holds for some A ⊂ X×G, x ∈ ΠnA, and ϕ
ny = ϕmx

for some y ∈ X and m ∈ N0. If n ≤ m, then y = ϕ
m−nx. Hence, if we take

j = m − n, i = n, and z = x, we obtain y = ϕm−nx ∈ ΠmA, by (3.1). If
n > m, then x = ϕn−my, and thus, ϕn−mx ∈ ΠnA. Hence y ∈ Πm, by (3.1).
Therefore A is saturated.

Assume now that A ⊂ X × G is saturated. Then, by Corollary 3.2, we have
ϕjΠiA ⊂ Πi+jA. Hence, if z ∈ Πi, then ϕ

jz ∈ Πi+jA. Now, conversely, if
ϕjz ∈ Πi+jA, then z ∈ Πi since (z,ϕ

i) ∼ (ϕjz,ϕi+j) and A is saturated. �

Corollary 3.7. If G = {ϕn : n ∈ N0}, then A ⊂ X × G is saturated if and
only if

Πj−1A = ϕ
−1ΠjA

for every j ∈ N.

Theorem 3.8. If X is a normal space, ϕ : X → X is a closed and continuous
injection, and G = {ϕn : n ∈ N0}, then B(X,G) is a Hausdorff space.

Proof. Consider two distinct points in B(X,G). Without loss of generality,
we can assume that they are represented by x

ϕn
and y

ϕn
for some x,y ∈ X

and n ∈ N. Then x 6= y and there exist open sets Un,Vn ⊂ X such that
x ∈ Un, y ∈ Vn, and Un ∩ Vn = ∅. Since ϕ is a closed injective map, ϕ(Un)
and ϕ(Vn) are disjoint closed sets. Whereas X is normal, there exist open sets
Un+1,Vn+1 ⊂ X such that

ϕ(Un) ⊂ Un+1, ϕ(Vn) ⊂ Vn+1, and Un+1 ∩ Vn+1 = ∅.

Similarly, by induction, we can construct open sets Un+k,Vn+k ⊂ X such that

ϕ(Un+k) ⊂ Un+k+1, ϕ(Vn+k) ⊂ Vn+k+1, and Un+k+1 ∩ Vn+k+1 = ∅,

for all k = 1, 2, . . . . Now, for m = n,n + 1,n + 2, . . . , we define open subsets
of X × G:

U′m =

m
⋃

j=0

(

ϕj−mUm
)

× {ϕj} and V ′m =

m
⋃

j=0

(

ϕj−mVm
)

× {ϕj}.

Note that U′n ⊂ U
′
n+1 ⊂ . . . , V

′
n ⊂ V

′
n+1 ⊂ . . . , and U

′
m∩V

′
m = ∅ for all m ≥ n.

Finally, let

U =

∞
⋃

m=n

U′m and V =

∞
⋃

m=n

V ′m.

Clearly, U and V are disjoint open subsets of X ×G such that (x,ϕn) ∈ U and
(y,ϕn) ∈ V . Since U and V are defined the same way, it suffices to show that



On the topology of generalized quotients 211

U is saturated. Note that

ΠjU =

∞
⋃

m=n

ϕj−mUm

if j = 0, . . . ,n, and

ΠjU =

∞
⋃

m=j

ϕj−mUm

if j > n. Since Πj−1U = ϕ
−1ΠjU for every j ∈ N, it follows that U is saturated

by Corollary 3.7. �

Corollary 3.9. If X is a compact Hausdorff space and G is generated by a
continuous injection, then B(X,G) is a Hausdorff space.

Now we consider the case when X has an algebraic structure, namely X
is a topological semigroup. A nonempty set X with an associative operation
(x,y) → xy from X × X into X is called a semigroup. If the topology of X
is Hausdorff and the semigroup operation is continuous (with respect to the
product topology on X × X), then X is called a topological semigroup. Our
main result follows from a theorem of Lawson and Madison (see Theorem 1.56
in [3]).

Theorem 3.10 (Lawson and Madison). Let S be a locally compact σ-compact
semigroup and let R be a closed congruence on S. Then S/R is a topological
semigroup.

An equivalence ∼ in a semigroup A is called a congruence if

a ∼ b implies ca ∼ cb for all c ∈ A.

If (X, ·) is a semigroup and G is a commutative semigroup of injective ho-
momorphisms on X, then X × G is a semigroup with respect to the binary
operation ∗ defined by

(x,ϕ) ∗ (y,ψ) = ((ψx) · (ϕy),ϕψ),

where x,y ∈ X and ϕ,ψ ∈ G.

Lemma 3.11. The equivalence ∼ in X × G defined by

(x,ϕ) ∼ (y,ψ) if ψx = ϕy

is a congruence with respect to ∗.

Proof. Let (x,ϕ), (y,ψ), (z,γ) ∈ X × G and (x,ϕ) ∼ (y,ψ). Then

(x,ϕ) ∗ (z,γ) = ((γx) · (ϕz),ϕγ) and (y,ψ) ∗ (z,γ) = ((γy) · (ψz),ψγ).

Since ψx = ϕy and G is commutative, we have

ψγ((γx) · (ϕz)) = (ψγγx) · (ψγϕz) = (ϕγγy) · (ϕγψz) = ϕγ((γy) · (ψz)),

which means (x,ϕ) ∗ (z,γ) ∼ (y,ψ) ∗ (z,γ). �



212 J. Burzyk, C. Ferens and P. Mikusiński

A relation ∼ in a topological space Y is called closed if {(a,b) ∈ Y ×Y : a ∼
b} is a closed subset of Y × Y with respect to the product topology.

Lemma 3.12. If X is Hausdorff, then ∼ is a closed relation in X × G.

Proof. We have to show that the set

R = {((x,ϕ), (y,ψ)) : (x,ϕ), (y,ψ) ∈ X × G and (x,ϕ) ∼ (y,ψ)}

is closed in (X × G) × (X × G). Consider ((x,ϕ), (y,ψ)) /∈ R. Then (x,ϕ) 6∼
(y,ψ) and hence ψx 6= ϕy. Since X is Hausdorff, there are open and disjoint
U,V ⊂ X such that ψx ∈ U and ϕy ∈ V . Then

(x,ϕ) × (y,ψ) ∈ (ψ−1(U) × {ϕ}) × (ϕ−1(V ) × {ψ}).

Clearly, (ψ−1(U) × {ϕ}) × (ϕ−1(V ) × {ψ}) is open and disjoint with R. �

In view of the above lemmas, the theorem of Lawson and Madison gives us
the following result.

Theorem 3.13. If X is a Hausdorff semigroup and (X ×G) is locally compact
σ-compact, then B(X,G) is Hausdorff.

References

[1] D. Bradshaw, M. Khosravi, H. M. Martin and P. Mikusiński, On Categorical and Topo-
logical Properties of Generalized Quotients, preprint.

[2] J. Burzyk and P. Mikusiński, A generalization of the construction of a field of quotients
with applications in analysis, Int. J. Math. Sci. 2 (2003), 229–236.

[3] J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of Topological Semigroups
(Marcel Dekker, New York, 1983).

[4] P. Mikusiński, Generalized Quotients with Applications in Analysis, Methods Appl.
Anal. 10 (2004), 377–386.

Received May 2007

Accepted December 2007

Józef Burzyk

Institute of Mathematics, Technical University of Silesia, Kaszubska 23, 44-100,
Gliwice, Poland

Cezary Ferens (c.ferens@wp.pl)
ul. Batorego 77/1, 43-100 Tychy, Poland

Piotr Mikusiński (piotrm@mail.ucf.edu)
Department of Mathematics, University of Central Florida, Orlando, FL 32816-
1364, USA