@
Appl. Gen. Topol. 18, no. 1 (2017), 53-59

doi:10.4995/agt.2017.4676

c© AGT, UPV, 2017

On quasi-orbital space

Hattab Hawete

Laboratoire des systèmes dynamiques et combinatoires, Faculté des Sciences de Sfax, Tunisia.

Umm Al Qura University Makkah KSA. (hattab.hawete@yahoo.fr)

Communicated by F. Balibrea

Abstract

Let G be a subgroup of the group Homeo(E) of homeomorphisms of
a Hausdorff topological space E. The class of an orbit O of G is the

union of all orbits having the same closure as O. We denote by E/G̃
the space of classes of orbits called quasi-orbit space. A space X is

called a quasi-orbital space if it is homeomorphic to E/G̃ where E is
a compact Hausdorff space. In this paper, we show that every infinite
second countable quasi-compact T0-space is the quotient of a quasi-
orbital space.

2010 MSC: 54F65; 54H20.

Keywords: homeomorphism; group; quasi-orbit space; quasi-orbital space.

1. Introduction

The standard setting for topological dynamics is a group of homeomorphisms
G on a compact Hausdorff space E [6]. This group induces an open equivalence
relation defined by the family of orbits (Gx = {gx : g ∈ G},x ∈ E). We
denote by E/G the orbit space equipped with the quotient topology. The
study of this space is difficult: just consider the example of a group generated
by an irrational rotation on the circle; indeed the orbit space does not verify
the weaker separation axioms, as the T0 separation axiom. For this reason
[8, 1, 2, 7] consider an intermediary quotient, called the quasi-orbit space.

The class of the orbit Gx is G̃x =
⋃

O=Gx

O. The family (G̃x,x ∈ E) deter-

mines an open equivalent relation on E [8]. Let E/G̃ the space of classes of

Received 09 February 2016 – Accepted 21 July 2016

http://dx.doi.org/10.4995/agt.2017.4676


H. Hawete

orbits equipped with the quotient topology. The space of classes of orbits is

called the quasi-orbit space. The space E/G̃ is a T0-space and its the universal
T0-space associated to the orbit space E/G as in Bourbaki [3, Exercice 27 page

I-104]. Let p : E → E/G̃ be the canonical projection. The map p is open. The
map ϕ : E/G → E/G̃ which associates to each orbit its class is an onto quasi-
homeomorphism1. Thus E/G̃ is a good representative of E/G. According to

[8, 1], the space E/G̃ keeps information on the initial dynamical system.

A space X is a quasi-orbital space if it is homeomorphic to a quasi-orbit E/G̃
where E is a compact Hausdorff space and G is a subgroup of homeomorphisms
of E.

In [1], the authors asked the following problem: under which conditions a
T0-space is quasi-orbital? In [2] the authors showed that a finite T0-space is
quasi-orbital. Note that, according to [1, Example 3.4], if X is a non quasi-
compact space then E is not in general compact.

In this paper we study this problem for an infinite T0-space. Our main result
is the following:

Theorem 1.1. Every second countable quasi-compact T0-space is the quotient
of a quasi-orbital space.

If E is a locally compact second countable topological space and G is a

subgroup of homeomorphisms of E then, according to [8, 7], E/G̃ satisfies the
following properties:

(1) E/G̃ is sober2;

(2) If G has a minimal set then, E/G̃ is quasi-compact.

In this paper, we show that if E is a locally compact topological space and

G is a subgroup of homeomorphisms of E then, if E/G̃ is quasi-compact then
it is quasi-orbital.

The paper consists of three sections. After introduction we will show some
properties of the quasi-orbital space. In section 3 we prove the main theorem.

2. Quasi-orbital spaces

In this section we study some properties of the quasi-orbital spaces.

Proposition 2.1. A closed subspace of a quasi-orbital space is quasi-orbital.

Proof. Let Y be a closed subset of a quasi-orbital space X. There exist a
compact and Haudorff space E and a subgroup G of Homeo(E) such that X

is homeomorphic to to the quasi-orbit space E/G̃; let ϕ such homeomorphism.
S = p−1(ϕ(Y )) is an invariant compact subset of E. We denote by H = G/S

1A continuous map f : X → Y between two topological spaces is called a quasi-
homeomorphism if the map which assigns to each open set V ⊂ Y the open set f−1(V )
is a bijective map.

2A space X is sober if every irreducible, nonempty, closed subset M of X has a unique

generic point m, i.e. M = {m}.

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On quasi-orbital space

the induced subgroup of G on S. Since S is an invariant subset of E, we have
for each x ∈ S, H(x) = G(x).

We will show that S/H̃ is homeomorphic to ϕ(Y ) and so to Y . Let f :

S/H̃ → ϕ(Y ) which maps any class of an orbit Hx to the class of the orbit
Gx. We will prove now that the bijective map f is a homeomorphism.

Let V be an open subset of ϕ(X), that means that V = U ∩ϕ(X) where U
is an open subset of E/G̃. So we have

p−1(V ) = p−1(U) ∩p−1(ϕ(X)) = p−1(U) ∩S

since p−1(U) is an open subset of E, p−1(V ) is an open subset of S. Thus V

is an open subset of S/H̃ and so f is a continuous map.

Let p1 : S → S/H̃ be the canonical projection and let V be an open subset
of S/H̃, that means that p−11 (V ) is an open subset of S and so there exists an
open subset U of E such that p−11 (V ) = U ∩S. We have

V = p(p−11 (V )) = p(U ∩S)

Since S is invariant, we deduce that

V = p(U) ∩p(S) = p(U) ∩ϕ(X)

The fact that p is an open map implies that V is an open subset of ϕ(X).
Therefore f is an open map.

Thus f is a homeomorphism and so Y is a quasi-orbital space. �

Example 2.2. This example shows that Proposition 2.1 minus the hypothesis
that Y is closed is false. Let f be an increasing homeomorphism of [0, 1]
without fixed point in ]0, 1[ such that f(0) = 0, f(1) = 1 and f( 1

2
) = 3

4
. Let

(an) be an increasing sequence such that a0 =
1
2

and converges to 5
8
. Let

(bn) be a decreasing sequence such that b0 =
3
4

and converges to 5
8
. Let g

be a homeomorphism of [0, 1] such that its support is
⋃

n≥0 f
n([an,bn]) and

g(fn(an+1)) = f
n(an+1). Let G be the group of homeomorphisms of [0, 1]

generated by f and g. Let X = [0, 1]/G̃ be the quasi-orbital space. The
subspace Y = X − p( 5

8
) is not closed. On the other hand Y can not be a

quasi-orbital space because it is irreducible without generic point [8, Lemma
2.2].

Proposition 2.3. Let X be a quasi-orbital space and R be an equivalence
relation on X which have a closed continuous cross-section s3. Then X/R is
quasi-orbital.

Proof. Since s is closed, s(X/R) is a closed subset of X and so, according to
Proposition 2.1, s(X/R) is quasi-orbital. Since s is closed and continuous, it
will be an embedding and so X/R is homeomorphic to s(X/R) which implies
that X/R is quasi-orbital. �

3According to [13], if X/R is a T1-space and zero-dimensional, then there exists a contin-

uous cross-section for R.

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H. Hawete

Remark 2.4. If an open equivalence relation R has a closed and continuous
cross-section, then X/R is a T0-space. Indeed, let a and b two elements of X/R

such that {a} = {b}. Since s is continuous and closed, s({a}) = s({a}) = {s(a)}
and s({b}) = s({b}) = {s(b)} and so {s(a)} = {s(b)}. The fact that X is a
T0-space implies that s(a) = s(b) and so a = b (s is injective). Therefore X/R
is a T0-space.

Proposition 2.5. Let (Xi, i ∈ I) be a family of quasi-orbital spaces. Then the
product

∏
i∈I

Xi is quasi-orbital.

Proof. For every i ∈ I, Xi is quasi-orbital, then there exist a compact space
Ei and a subgroup Gi of Homeo(Ei) such that Xi is homeomorphic to the

quasi-orbits space Ei/G̃i. Let E =
∏
i∈I

Ei be the product space and G =
∏
i∈I

Gi

be the product group. By applying [3, Proposition 7 TG I.27], we have, for

each x = (xi, i ∈ I), G(x) =
∏
i∈I

Gi(xi) and so G̃ =
∏̃

i∈I Gi =
∏

i∈I G̃i. By

applying [3, Corollaire p.TG I.34] it follows that
∏
i∈I

Xi is homeomorphic to

E/G̃. Since E is compact,
∏
i∈I

Xi is quasi-orbital. �

Proposition 2.6. If E is a locally compact space and G is a subgroup of

homeomorphisms of E, then if E/G̃ is quasi-compact then it is a quasi-orbital
space.

Proof. Since E/G̃ is a quasi-compact space, according to [7, Proposition 2.1],
G has a minimal set M. The fact that E − M is an open set of a locally
compact set implies that E −M is a locally compact space [3, Proposition 13
TG I.66]. we denote by H = G/E −M the induced subgroup of G on E −M.
Since E − M is invariant, we have for each x ∈ E − M, H(x) = G(x). Let
Ê = (E − M) ∪{ω} be the one point compactification of E − M. We can
suppose that H is a group of homeomorphisms of Ê by putting H(ω) = {ω}.

It is easy to see that the bijection f : Ê/H̃ → E/G̃ which maps any class of
an orbit Hx to the class of the orbit Gx for all x ∈ E −M and f(ω) = p(M)
is a homeomorphism. Thus E/G̃ is homeomorphic to Ê/H̃. �

3. Proof of Main Theorem

Recall that, a topological space X is a k-space (compactly generated) if the
following holds: a subset A ⊂ X is closed in X if and only if A∩K is closed in K
for every compact subset K ⊂ X [10]. It is easy to see that the family of closed
compact sets determines the topology of a k-space. Any locally compact space
is a k-space and any first countable topological space (in particular a metric
space) is a k-space. According to [4, p. 248], X is a k-space if and only if it is a
quotient space of a locally compact space Z. The space Z is a disjoint sum of

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On quasi-orbital space

all compact subsets (Ki, i ∈ I) of X: Z =
∐
i∈I

Ki = {(x,i) : i ∈ I and x ∈ Ki}.

The equivalence relation R on Z is defined by: (x,i)R(y,j) if x = y. Note that
Z is equipped with the disjoint sum topology defined by: U is an open set of
Z if ϕ−1j (U) is an open set of Kj where the map ϕj : Kj → Z is defined by
ϕj(x) = (x,j). Recall that, for all j, the map ϕj is continuous closed and open
and f : Z → Y is continuous if and only if f ◦ϕj is continuous.

Remark 3.1. The set S = {0, 1} equipped with the topology {∅,S,{1}} is
called the Sierpinski space; it is a connected T0-space but it is not a T1-space.
If G1 is a finitely generated abelian subgroup of Diff

∞
+ (S

1) of finite rank k ≥ 2
having only a one fixed point e ∈ S1, then all other orbits are everywhere
dense (N. Kopell, G. Reeb [11], [12]). Thus the quasi-orbits space S1/G̃1 is
homeomorphic to the Sierpinski space S.

Proof (Main Theorem). Since X is a T0-space, by applying [5, Theorem 2.3.26
p.84], there exists an embedding ψ : X →

∏
i∈I Si (where Si is the Sierpinski

space {0, 1}). We can suppose that I ⊂ N; indeed, X is second countable. We
know that for each i ∈ I there is a homeomorphism fi : Si → S1i /G̃i where S

1
i is

the unit circle S1 and Gi is the group G1 defined in Remark 3.1. The product
map

∏
i∈I fi :

∏
i∈I Si →

∏
i∈I S

1
i /G̃i is also a homeomorphism. According to

[3, Corollaire p.TG I.34],
∏

i∈I S
1
i /G̃i is homeomorphic to

∏
i∈I S

1
i /

∏
i∈I G̃i.

The space TI =
∏

i∈I S
1
i is a compact second countable metric space. We

put GI =
∏

i∈I Gi. The group G
I is abelian. Then we conclude that there

exists an embedding ϕ : X → TI/G̃I . Let p : TI → TI/G̃I be the canonical
projection. We denote by E = p−1(ϕ(X)) and we denote by G = GI/E the
induced subgroup of GI on E. Since E is a saturated subset of TI . We have
for each x ∈ E, G(x) = GI (x).

We will show that E/G̃ is homeomorphic to ϕ(X) and so to X. Let f :

E/G̃ → ϕ(X) ⊂ TI/G̃I which maps any class of an orbit G(x) to the class of the
orbit GI (x). We will prove now that this bijective map f is a homeomorphism:

Let V be an open subset of ϕ(X), that means that V = U ∩ϕ(X) where U
is an open subset of TI/G̃I . So we have

p−1(V ) = p−1(U) ∩p−1(ϕ(X)) = p−1(U) ∩E

since p−1(U) is an open subset of TI , p−1(V ) is an open subset of E. Thus V
is an open subset of E/G̃ and so f is a continuous map.

Let p1 : E → E/G̃ be the canonical projection and let V be an open subset
of E/G̃, that means that p−11 (V ) is an open subset of E and so there exists an
open subset U of TI such that p−11 (V ) = U ∩E. We have

V = p(p−11 (V )) = p(U ∩E)

Since E is saturated, we deduce that

V = p(U) ∩p(E) = p(U) ∩ϕ(X)

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H. Hawete

The fact that p is an open map implies that V is an open subset of ϕ(X).
Therefore f is an open map. We conclude that f is a homeomorphism.

Since E is a metric space, it is first countable and so E is a k-space. Thus
E is the quotient of a locally compact metric space F by the relation R. Note
that F is the disjoint union of all compact subsets of E. Let q : F → F/R = E
be the canonical projection.

Let g be an element of G. We define on F the map g : F → F by g(x,i) =
(g(x),j) where g(Ki) is the compact Kj. It is easy to see that g is a well defined

bijection. Let U be an open set of F , then U =
∐
i∈I

Ui ∩Ki where Ui is an open

set of E. g−1(U) =
∐
i∈I

g−1(Ui) ∩g−1(Ki) and g(U) =
∐
i∈I

g(Ui) ∩g(Ki) and

since g is a homeomorphism g(Ui) and g
−1(Ui) are open sets of E and g is a

permutation of the set of all compact subsets. Then g−1(U) =
∐
i∈I

g−1(Ui) ∩Ki

and g(U) =
∐
i∈I

g(Ui) ∩Ki are open sets of F. Therefore g is a homeomorphism

of F. The set G = {g : g ∈ G} is a subgroup of homeomorphisms of F .
Since E/G̃ is quasi-compact, we show Now that G has a minimal set. We

start by showing that E/G̃ contains a point a such that {a} is closed. Since E/G̃
is quasi-compact, by Zorn’s lemma, it contains a minimal set M. Therefore

for all z ∈ M we have {z} = M. From the fact that E/G̃ is a T0-space, it
follows that M is a single point set {a} (indeed {a} = {b} ⇒ a = b). Let x be
an element of E such that p(x) = a. The fact that {a} is closed implies that
p−1({a}) = G̃x is a closed invariant set of E such that if y ∈ G̃x then Gy = Gx
and so G̃x is a minimal set of G. q−1(G̃x) is a closed subset of F . If there exist

(x,i) ∈ q−1(G̃x) and g ∈ G such that g(x,i) = (g(x),j) is not in q−1(G̃x),
then q(g(x),j)) is not in G̃x and so g(x) is not in G̃x which contradicts the

fact that G̃x is an invariant set. We conclude that q−1(G̃x) is a minimal set of
G.

The fact that F is locally compact, according to [7, Proposition 2.1], implies

that F/G̃ is quasi-compact. Then, by applying Proposition 2.6, we have F/G̃

is a quasi-orbital space Ê/H̃. Let h be the homeomorphism of Ê/H̃ and F/G̃.

Let p2 : F → F/G̃ and p3 : E → E/G̃ be the canonical projections. Let
q̃ : F/G̃ → E/G̃ be the map defined by q̃ ◦p2 = p3 ◦ q. q̃ is a continuous and
onto map. The map q̂ = ϕ−1 ◦f ◦ q̃ ◦h is a continuous and onto map of Ê/H̃
to X which implies that X is a quotient of a quasi-orbital space. 2

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 58



On quasi-orbital space

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c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 59