()


@ Appl. Gen. Topol. 19, no. 1 (2018), 1-7doi:10.4995/agt.2018.7012
c© AGT, UPV, 2018

Partially topological group action

M. A. Al Shumrani

Department of Mathematics, King Abdulaziz University, P.O.Box: 80203 Jeddah 21589, Saudi

Arabia (maalshmrani1@kau.edu.sa)

Communicated by S. Garćıa-Ferreira

Abstract

The concept of partially topological group was recently introduced in

[3]. In this article, we define partially topological group action on par-
tially topological space and we generalize some fundamental results

from topological group action theory.

2010MSC:Primary: 54A25; Secondary: 54B05.

Keywords: partially topological space; partially topological group; group

action.

1. Partially topological spaces

In this section, we recall definition of the categoryGTSpt of partially topo-
logical spaces and strictly continuous mappings which was defined in [4].

Definition 1.1. Let X be any set, τX be a topology on X. A family of open
families CovX ⊆ P(τX) will be called a partial topology if the following
conditions are satisfied:

(i) if U ⊆ τX and U is finite, then U ∈CovX;
(ii) if U ∈CovX and V ∈ τX, then {U∩V :U ∈U}∈CovX;
(iii) if U ∈ CovX and, for each U ∈ U, we have V(U) ∈ CovX such that
⋃

V(U)=U, then
⋃

U∈U
V(U)∈CovX;

(iv) ifU ⊆ τX andV ∈CovX are such that
⋃

V =
⋃

U and, for eachV ∈V
there existsU ∈U such that V ⊆U, then U ∈CovX.

Received 09 December 2016 – Accepted 18 September 2017

http://dx.doi.org/10.4995/agt.2018.7012


M. A. Al Shumrani

Elementsof τX are calledopen sets, andelements ofCovX are calledadmissi-
ble families.We say that (X,CovX) is apartially topological generalized
topological space or simply partially topological space. For simplicity,
from now on, we shall denote a partially topological space (X,CovX) byX.
LetX andY bepartially topological spaces and let f :X →Y bea function.

Then f is called strictly continuous if f−1(U)∈CovX for anyU ∈CovY . A
bijection f :X →Y is called a strictly homeomorphism if both f and f−1

are strictly continuous functions. Ifwe have a strictly homeomorphismbetween
X and Y we say that they are strictly homeomorphic and we denote that
byX ∼=Y .

Remark 1.2. Theabovenotion of partial topology is a special case of the notion
of generalized topology in the sense of H. Delfs and M. Knebusch considered
in [2, 4, 5, 6, 7], when the family OpX of open sets of the generalized topology
forms a topology.

Definition 1.3. Let (X,CovX) be a partially topological space and let Y be
a subset ofX. Then the partial topology

CovY =(〈CovX ∩2Y 〉Y )pt,

that is: the smallestpartial topologycontainingCovX∩2Y , is calledasubspace
partial topology onY , and (Y,CovY ) is a subspace of (X,CovX). (It is also
the smallest generalized topology containing CovX ∩2Y .)

Fact 1.4. Let ϕ : X → X′ be a mapping between partially topological spaces
and let Y be a subspace of X. Then the following are equivalent:

a) ϕ is strictly continuous,
b) the restriction mapping ϕ|Y :Y →X

′ is strictly continuous.

Definition 1.5. Let (X,CovX) and (Y,CovY ) be two partially topological
spaces. The product partial topology on X × Y is the partial topology
CovX×Y =(〈CovX ×2CovY 〉X×Y )pt in the notation of Definition 4.6 of [7]; in
otherwords: the smallestpartial topology inX×Y that containsCovX×2CovY .

Recall that amapping f :X →Y is said to be an openmapping if for every
open setU of X, the set f(U) is open in Y . It is said to be a closed mapping
if for every closed set A of X, the set f(A) is closed in Y . Also, recall that a
surjective mapping f : X → Y is said to be a quotient mapping provided a
subsetU of Y is open in Y if and only if f−1(U) is open inX.

2. Partially Topological Groups

In this section, we recall the definition of partially topological group. This
notion was recently introduced in [3].

Definition 2.1. Apartially topological groupG is an ordered pair ((G,∗),
CovG) such that (G,∗) is a group, while CovG is a generalized topology onG
such that

⋃

CovG is a T1 topology on G and the multiplication mapping of
(G×G,CovG×G) into (G,CovG), which sends ordered pair (x,y) ∈ G×G

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 2



Partially topological group action

to x ∗ y, is strictly continuous and the inverse mapping from (G,CovG) into
(G,CovG),which sends eachx∈G tox

−1, is strictly continuous.For simplicity,
from now on, we shall denote a partially topological group ((G,∗),CovG) by
G.

Definition 2.2. Any subgroupH of a partially topological groupG is a par-
tially topological group and it is called apartially topological subgroup of
G.

Definition 2.3. Letϕ :G→G′ be a function. Thenϕ is called amorphism
ofpartially topological groups ifϕ is both strictly continuousandgroupho-
momorphism.Moreover,ϕ is an isomorphism if it is strictly homeomorphism
and group isomorphism.
If we have an isomorphism between two partially topological groupsG and

G′, then we say that they are isomorphic and we denote that byG∼=G′.

Remark 2.4. Obviously composition of twomorphisms of partially topological
groups is amorphism. In addition, the identitymapping is an isomorphism. So
partially topological groups and their morphisms form a categoryPTGr.

3. Partially Topological Group Action On Partially
Topological Space

In this section, we introduce partially topological group action on partially
topological space andwe extend some fundamental results in [1] of action of a
topological group on a topological space to this new concept.

Definition 3.1. IfG is a partially topological groupwith identity e andX is a
partially topological space, then anaction ofGonX is amappingG×X →X,
with the image of (g,x) being denoted by g(x), such that (gh)(x) = g(h(x))
and e(x)=x for all g,h∈G and x∈X.

If this mapping is strictly continuous, then the action is said to be strictly
continuous.
The space X, with a given strictly continuous action of G on X, is called

partially G-space.
For a point x∈X, the setG(x)= {gx : g∈G} is called the orbit of x.

Definition 3.2. Let G be a partially topological group and X a partially
topological space. LetG act onX. For a point x ofX, the set

Gx = {g∈G : gx=x} (or Gx = {g∈G :xg=x})

is called the stabilizer of x.

Fact 3.3. The stabilizer Gx of any point x∈X is a subgroup of G.

Definition 3.4. Let G be a partially topological group and X a partially
topological space. LetG act onX. For a point x ofX, we define amapping

µx :G→X

by µx(g)= gx (or µx(g)=xg).

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 3



M. A. Al Shumrani

Note that µx is strictly continuous by strictly continuity of the action. The
action is called transitive if for each x∈X,Gx =X. Then Obviously we have
the following fact.

Fact 3.5. µx is surjective iff G acts transitively on X.

Proposition3.6. Every strictly continuous action θ :G×X →X of a partially
topological group G on a partially topological space X is an open mapping.

Proof. It suffices to prove that the images under θ of the elements of some base
for G×X are open in X. LetO=U ×V ⊂G×X, whereU and V are open

sets inG andX, respectively. Then θ(O) =
⋃

g∈G

θg(V ) is open inX since every

θg is a strictly homeomorphism of X onto itself. Since the open sets U ×V
form a base forG×X, the mapping θ is open. �

Proposition 3.7. The strictly continuity of an action θ : G×X → X of a
partially topological groupGwith identity e on a partially topological spaceX is
equivalent to the strictly continuity of θ at the points of the set {e}×X ⊂G×X.

Proof. Let g ∈ G and x ∈ X be arbitrary and U be a neighborhood of gx in
X. Since θh is a homeomorphism of X for each h ∈ G, the set V = θg−1(U)
is a neighborhood of x in X. By the strictly continuity of θ at (e,x), we can
find a neighborhood O of e in G and a neighborhood W of x in X such that
hy ∈ V for all h ∈ O and y ∈ W . Clearly, if h ∈ O and y ∈ W , then
(gh)(y) = g(hy) ∈ gV = θg(V ) = U. Thus, ky ∈ U, for all k ∈ gO and all
y ∈ W , where O′ = gO is a neighborhood of g in G. Hence, the action θ is
strictly continuous. �

Next we present two examples of strictly continuous actions of partially
topological groups.

Example 3.8. Any partially topological groupG acts on itself by left transla-
tions, that is, θ(x,y)=xy for allx,y∈G. The strictly continuity of this action
follows from the strictly continuity of the multiplication inG.

Example 3.9. LetG be a partially topological group,H a closed subgroup of
G, and let G/H be the corresponding left coset space. The action φ of G on
G/H, defined by the rule φ(g,xH) = gxH, is strictly continuous. Indeed, let
y0 ∈G/H, and fix an open neighborhoodO of y0 inG/H. Choosex0 ∈G such
that π(x0) = y0, where π : G → G/H is the quotient mapping. There exist
open neighborhoodsU and V of the identity e inG such that π(Ux0)⊂O and
V 2 ⊂U. Clearly, W = π(Vx0) is open in G/H and y0 ∈W. By the choice of
U and V , if g ∈ V and y ∈ W , then φ(g,y) ∈ O. Indeed, let x1 ∈ Vx0 with
π(x1) = y. Then y = x1H and φ(g,y) = gx1H ∈ VVx0H ⊂ π(Ux0) ⊂ O.
Therefore,φ is continuous at (e,y0)∈G×G/H. Hence,φ is strictly continuous
by Proposition 3.7.

Suppose that a partially topological groupG acts strictly continuously on a
partially topological spaceX and thatX/G is the corresponding orbit set. Let

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 4



Partially topological group action

X/G have the partially quotient topology generated by the orbital projection
π : X → X/G (a subset U ⊂ X/G is open in X/G if and only if π−1(U) is
open in X). The partially topological space X/G is called the orbit space or
the orbit space of the partillayG-spaceX.
The following result shows that the orbital projection is always an open

mapping.

Proposition 3.10. If θ : G×X → X is a strictly continuous action of a
partially topological groupG on a partially topological spaceX, then the orbital
projection π :X →X/G is an open mapping.

Proof. For any open set U ⊂ X, consider the set π−1π(U) = GU. Every
left translation θg is a strictly homeomorphism of X onto itself, so the set

GU =
⋃

g∈G

θg(U) is open in X. Since π is a quotient mapping, π(U) is open in

X/G. Hence, π is an openmapping. �

Theorem 3.11. Suppose a compact partially topological group H acts strictly
continuously on a Hausdorff partially space X, then the orbital projection π :
X →X/H is both open and perfect mapping.

Proof. First note that π is open by Proposition 3.10. Next we show that π is
perfect. Let y ∈X/H, choose x∈X such that π(x) = y. Note that π−1(y) =
Hx is the orbit of x inX. Since themapping ofH ontoHx assigning to every
g ∈H the point gx∈X is strictly continuous, the image Hx of the compact
groupH is also compact. Hence, all fibers of π are compact.
We show that the mapping π is closed. Let y ∈ X/H and x ∈ X such

that π(x) = y. Let O be an open set in X containing π−1(y) = Hx. Since
the action of H on X is strict continuous, we can find, for every g ∈ H,
open neighborhoods g ∈ Ug and x ∈ Vg in H and X, respectively, such that
UgVg ⊂O. By the compactness ofH and of the orbitHx, there exists a finite

set F ⊂H such that H =
⋃

g∈F

Ug and Hx⊂
⋃

g∈F

gVg. Then V =
⋂

g∈F

Vg is an

open neighborhood of x in X, and we claim that HV ⊂ O. Indeed, if h ∈ H
and z ∈ V , then h ∈ Ug, for some g ∈ F , so that hz ∈ UgV ⊂ UgVg ⊂ O.
Thus, W = π(V ) is an open neighborhood of y in X/H, and we have that
π−1π(V )=HV ⊂O. Hence, π is closed. �

Definition 3.12. LetX and Y be partiallyG-spaces with strictly continuous
actions θX :G×X →X and θY :G×Y →Y.A strictly continuous mapping
f : X → Y is called partially G-equivariant if θY (g,f(x)) = f(θX(g,x)),
that is, gf(x) = f(gx), for all g ∈ G and all x ∈ X. Clearly, f is partially
G-equivariant if and only if the following diagram

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 5



M. A. Al Shumrani

G×X
θX
−−−−→ X





y

F





y

f

G×Y
θY
−−−−→ Y

commutes, where F = idG × f is the product of the identity mapping idG of
G and the mapping f.

Example 3.13. Let H be a closed subgroup of a partially topological group
G, and Y = G/H be the left coset space. Denote by θG the action of G on
itself by left translations, and by θY the natural strictly continuous action of
G on Y . Then the quotient mapping π :G→G/H defined by π(x) = xH for
each x∈G is equivariant. Indeed, the equality g(π(x)) = gxH = π(gx) holds
for all g,x∈G. Equivalently, the following diagram

G×G
θG
−−−−→ G





y

Π





y

π

G×Y
θY
−−−−→ Y

commutes, where Π= idG×π.

Let η = {Xi : i ∈ I} be a family of partially G-spaces. Then the product
spaceX =

∏

i∈I
Xi, ifX isHausdorff, is apartiallyG-space.Todefineanaction

of G on X, take any g ∈G and any x= (xi)i∈I ∈X, and put gx= (gxi)i∈I.
Thus,G acts onX coordinatewise.
The following result shows the strictly continuity of this action.

Proposition3.14. The coordinatewise action ofG on the productX =
∏

i∈I
Xi

of partially G-spaces is strictly continuous, that is, X is a partially G-space, if
X is Hausdorff.

Proof. ByProposition 3.7, it suffices to verify the continuity of the action ofG
onX at the neutral element e∈G. Let x=(xi)i∈I ∈X be an arbitrary point
andO⊂X aneighborhoodof gx inX. Since canonical open sets formabase of
X, we can assume that O=

∏

i∈I
Oi, where eachOi is an open neighborhood

of xi in Xi and the set F = {i ∈ I : Oi 6= Xi} is finite. Since all factors are
partiallyG-spaces, we can choose, for every i∈F , open neighborhoods e∈Ui
and xi ∈ Vi in G and Xi, respectively, such that UiVi ⊂ Oi. Put U =

⋃

i∈F

Vi

andW =
∏

i∈I
Wi, whereWi =Vi if i∈F andWi =Xi otherwise. Therefore,

it follows from the definition of the sets U and W that UW ⊂ O. Hence, the
action ofG onX is strictly continuous. �

Theorem 3.15. LetG be a partially topological group andX a partially topo-
logical space. Let G act on X. Suppose that both G and X/G are connected,
then X is connected.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 6



Partially topological group action

Proof. Suppose that X is the union of two disjoint nonempty open subsets
U and V . Now π(U) and π(V ) are open in X/G. Since X/G is connected,
π(U) and π(V ) cannot be disjoint. If π(x)∈π(U)∪π(V ), then bothU∪O(x)
and V ∪O(x) are nonempty, whereO(x) is the orbit of x. It means O(x) is a
disjoint union of two nonempty open sets. But O(x) is the image of G under
the strictly continuous function f :G→X defined by f(g)= g(x). Therefore,
O(x) is connected which is a contradiction. Hence,X is connected. �

Theorem 3.16. If X is a compact partially topological group and G a closed
subgroup acting on X by left translation, then X/G is regular.

Proof. SinceG is closed subgroupandthe left translationmappingLx :X →X
is strictly homeomorphism then π−1π(x) = xG=Lx(G) is closed. Thus every
point π(x) ofX/G is closed, and it follows thatX/G is T1 space.
Nowwe show that for a closed subsetF ofX/G and a point p /∈F there are

open sets U,V satisfying p ∈ U,F ⊂ V,U ∩V = ∅. Since X acts transitively
on X/G, we may assume that p is an element of the class eG = G of the
identity element e. Since F is closed, there exists an open set U0 such that
F ∩U0 = ∅ and p ∈ U0. From the strictly continuity of group action of
X, there is an open set W such that e ∈ W and W−1W ⊂ π−1(U0). The set

Wπ−1(F)=
⋃

x∈π−1(F)

Wx is open. Sinceπ is an openmapping, bothU =π(W)

and V =π(Wπ−1(F)) are open sets and p∈U and F ⊂V .
Next we show thatU ∩V = ∅. Suppose that there exists y∈U∩V . Then

there existx1,x2 ∈W andx∈π
−1(F) such that y=π(x2)=π(x1x).Thus,we

haveg∈G suchthatx2g=x1x, fromwhichwededuce thatπ(xg
−1)∈F∩U0 =

∅ from xg−1 =x1
−1x2 ∈W

−1W ⊂π−1(U0). Therefore,U∩V = ∅. �

References

[1] A. Arhangel’skii and M. Tkachenko, Topological groups and related structures, World
Scientific, 2008.

[2] H. Delfs and M. Knebusch, Locally semialgebraic spaces, Lecture Notes in Math. 1173,
Springer, Berlin-Heidelberg, 1985.

[3] C.Ozel, A. Piękosz,M.A.Al Shumrani andE.Wajch, Partially paratopological groups,
Topology Appl. 228 (2017), 68–78.

[4] A. Piękosz, On generalized topological spaces I, Ann. Polon. Math. 107, no. 3 (2013),
217–241.

[5] A. Piękosz, On generalized topological spaces II, Ann. Polon. Math. 108, no. 2 (2013),
185–214.

[6] A. Piękosz, O-minimal homotopy and generalized (co)homology, Rocky Mountain J.
Math. 43, no. 2 (2013), 573–617.

[7] A. Piękosz and E. Wajch, Compactness and compactifications in generalized topology,
Topology Appl. 194 (2015), 241–268.

c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 1 7