SQU Journal for Science, 2022, 27(2),100-106  DOI:10.53539/squjs.vol27iss2pp100-106 

Sultan Qaboos University  

100 

 

Topo-Groups and aTychonoff Type Theorem 

Mohammad Shahryari
 

Department of Mathematics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-
Khod, PC 123, Muscat, Sultanate of Oman, Email: m.ghalehlar@squ.edu.om. 

ABSTRACT: We define a topo-system on a group as a set of subgroups satisfying certain topology-like conditions. 

We study some fundamental properties of groups equipped with a topo-system (topo-groups) and using a suitable 

formalization of a subgroup filter, we prove a Tychonoff type theorem for the direct product of topo-compact topo-

groups. 

 

Keywords: Topo-groups; Topo-systems; Filters of subgroups; Subgroup ultra-filter  

 ونظرية من صنف تيخونوف ةفضائي -زمر

 محمد شهرياري 

سوف نقوم بدراسة بعض  محددة.فضائية على الزمرة بأنه مجموعة من الزمر الجزئية التي تحقق شروط شبه  فضائي يتم تعريف نظام :صلخمال

الزمر الجزئية، سوف نبرهن نظرية من صنف  فالتر( و باستعمال تشكيالت مناسبة من فضائية -)زمر فضائيالتي تملك نظام للزمر ساسيةالخصائص األ

 فضائياً.  المتراصة الفضائيةباالستنساخ الضربي للزمر  تيخونوف والتي تتعلق

 

 .فضائية، فالتر الزمر الجزئية، فالتر فائقة للزمر الجزئية نظم ،زمرفضائية :مفتاحيةالكلمات ال
 

 

1. Introduction 

In this article, we introduce an interesting topology-like concept concerning groups (which, by almost the same 

method, can be defined for other algebraic systems, see [1]). Given an arbitrary group G, we define a topo-system on 

G, as a set of subgroups satisfying certain conditions like a topology on a set. We will call such a group a topo-group. 

These topo-groups are not rare and as we will see, there are many examples of topo-groups. We investigate 

fundamental notions concerning topo-groups and we see that many basic concepts of topology can be introduced in 

the frame of topo-groups. A filter of subgroups of a group G will be defined in such a way that we will be able to 

formulate a Tychonoff type theorem for the direct product of topo-compact topo-groups. Throughout this article, 

Sub(G) and Sub∗(G) will be used for the set of subgroups and the set of non-identity subgroups of G, respectively. 

Also, for a set of subgroups S={Ai}i∈I, the notation <Ai : i ∈ I> represents the subgroup generated by S. For basic 
concepts of group theory, the reader can see [2].  

 

Definition 1.1. Let G be a group and T ⊆ Sub(G) be a set satisfying: 

a) 1 and G ∈ T, 
b) if {Ai}i∈I is a family of elements of T, then < Ai : i ∈ I > ∈ T, and  
c) if A and B are elements of T, then A ∩ B ∈ T. 

 

Then we call T a topo-system on G and the pair (G,T) a topo-group. Elements of T will be called T-open 

subgroups. 

 

Example 1.2. It is quite easy to find many examples of topo-groups. Here, we give a list of such examples: 

1- The set T = Sub(G) is a topo-system and we call it the discrete topo-system on G. 
2- The set T = {1, G} is a topo-system and we call it the trivial topo-system. 



TOPO-GROUPS AND A TYCHONOFF TYPE THEOREM  

101 

 

3- Let B ⊆ G and TB = {A ≤ G : B ⊆ A} ∪ {1}. This is the principal topo-system associated with B. 
4- Let Tcf = {A ≤ G :  [G : A] < ∞} ∪  {1}, where [G : A] is the index of A inside G. This is the cofinite topo-

system on G. 

5- Let Tn = {A ≤ G : A is normal in G}. This is the normal topo-system on G. 
6- Let Tchar = {A ≤ G : A is characteristic in G}. This is the characteristic topo-system on G. Recall that A is a 

characteristic subgroup of G if it is invariant under every automorphism of G. 

7- Let G be a topological group and T be the set of all open subgroups of G together with the identity subgroup. 
Then T is a topo-system on G. 

8- Let V be a variety of groups and TV = {A  ∈ G : G/A ∈ V} ∪  {1}. Then TV is a topo-system on G and, in fact, we 
have TV = TH ∩ Tn, where H = O

V
(G) is the V-residual of G (i.e., the smallest normal subgroup of G, such that G/H 

belongs to V). 
 

Definition 1.3. A subgroup A ≤ G is T-closed if for all x ∈ G \ A, there exists B ∈ T such that x ∈ B and A ∩ B = 1. 
It is easy to verify that 1 and G are T-closed, and the intersection of any family of T-closed subgroups is T-closed. 

Let X ≤ G and x ∈ X. Suppose there is a T-open A with x ∈ A ≤ X. Then we say that x is an interior element of X. The 
set of all interior elements of X is denoted by X

◦ 
and it is called the interior of X. Clearly, the interior of X is a T-open 

subgroup. As an example, in the case of the normal topo-system (see the above example), we have X
◦ 
= XG, the core of 

X (i.e., the maximal normal subgroup of G included in X). In the case of the co-finite topo-system, it can be verified 

that X
◦ 
= 1 or X. For any subgroup X, the boundary is ∂X = X \ X

◦
. A point x ∈ G is a limit point of X, if for all T-open 

A containing x, we have |A ∩ X| ≥ 2. The T-closure of X is the subgroup generated by X and its limit points. Clearly,  

it is T-closed. 

Unlike the ordinary topology, there is no symmetry between the concepts of T-open and T-closed subgroups. For 

example, if any subgroup of a topo-group G is T-open, we cannot say that any subgroup is T-closed as well. 

For example, consider the group 𝐺 = 𝑍 × 𝑍4  with the discrete topo-system. Then the subgroup A = <(0,2)> is not T-
closed. In general, we have the following proposition. 

 

Proposition 1.4. Let all cyclic subgroups of G be T-closed. Then all non-identity elements of G have prime orders. 

Proof. Let x ∈ G be a non-identity element. Then A = <x
2
> is T-closed. If x does not belong to A, then there is a B ∈ T 

containing x such that A ∩ B = 1. So, <x> ∩ A = 1 and hence x
2 

= 1. If x ∈ A, then x has finite order n. Suppose n = 
ab. Then <x

a
> is T-closed and hence 

x ∈ <x
a
>  or  <x

a
> ∩ <x> = 1.                                                                (1) 

 

In the first case we have x
b 
= 1 and in the second case, we have x

a 
= 1. So, n is a prime.  

 

Definition 1.5. A subgroup X ≤ G is a topo-compact subgroup if any T-open covering of X has a finite sub-covering, 

i.e., if X ⊆ ∪ i∈IAi, with Ai ∈ T, then there is a finite subset J ⊆ I, such that X ⊆ ∪ i∈JAi. In the special case X = G, we 
say that G is topo-compact. 

For example, a group is topo-compact with respect to the discrete topo-system if it is a finite union of cyclic 

subgroups. In fact, such a group is finite or infinite cyclic as the next proposition shows. 

 

Proposition 1.6. Let (G,T) be topo-compact and suppose that T is the discrete topo-system. Then G is finite or infinite 

cyclic. 

 

Proof. We have G = ∪ x∈G<x> and all <x> are T-open. So, by topo-compactness, 
 

𝐺 =< 𝑥1 >∪< 𝑥2 >∪ ⋯ ∪< 𝑥𝑛 > ,                                                       (2) 
 

for some finite set of elements.  Now by a theorem of Neumann (see [2], page 120), we may assume that any <xi> has a 

finite index. Suppose G is infinite. Then all <xi> are infinite as they have finite index, and hence G is torsion free. 

Assume H ≤ G to be a non-identity subgroup and  h ∈ H  to be a non-identity element. Then, there is an index i such 
that h ∈ <xi> and therefore [<xi> : <h>] is finite. So, already [G : <h>] is finite and hence H has finite index. A theorem 
of Fedorov (see [3], page 446) says that if any non-trivial subgroup of an infinite group has finite index, then the group 

is isomorphic to the infinite cyclic group Z. This completes the proof. 
The concept of topo-compact groups may have many interesting interpretations if we consider different topo-

systems. It can be verified that a T-closed subgroup of a topo-compact group is a topo-compact subgroup. 

If (G,T) is a topo-group and H ≤ G, then the set S = {A∩H : A ∈ T} generates a topo-system on H, which is an 
induced topo-system on H. We denote this topo-system by Tind. We give a recursive construction of the elements of 

Tind. Let T0 = S and for any ordinal α (see [4]) 

 
Also for limit ordinals we set 



MOHAMMAD SHAHRYARI
 

102 

 

𝑇α = ⋃ 𝑇β
β<α

 

 

Then it can be shown that Tind = ∪ αTα. Now, if (H,Tind) is a topo-compact topo-group, then it can be proved that H is a 
topo-compact subgroup of G. The converse situation is very complicated because the T-open subgroups of (H,Tind) 

have very complex structures. 

Also if H∈ G, then one checks that the set 

 
is a topo-system on the quotient group G/H. So, we call it the quotient topo-system. If {(Gi,Ti)i∈I} is a family of topo-

groups and 𝐺 = ∏ 𝐺𝑖𝑖 , then the set 

{∏ 𝐴𝑖
𝑖

:  𝐴𝑖 ≤ 𝐺𝑖 , 𝑎𝑛𝑑 𝑎𝑙𝑚𝑜𝑠𝑡 𝑎𝑙𝑙 𝐴𝑖 = 𝐺𝑖 } 

is a topo-system on G. Note that, the situation here is much better than the case of ordinary product topology because 

of the following two trivial equalities: 

1. (∏ 𝑨𝒊𝒊 ) ∩ (∏ 𝑩𝒊𝒊 ) = ∏ (𝑨𝒊 ∩ 𝑩𝒊)𝒊  

2. < ∏ 𝑨𝒊𝒋𝒊 :  𝒋 ∈ 𝑱 >= ∏ < 𝑨𝒊𝒋:  𝒋 ∈ 𝑱𝒊 > 

 

Definition 1.7. Let (G,T) and (H,S) be two topo-groups. A continuous map (or a topomorphism) is any 

homomorphism f : G → H with the property f
−1

(B) ∈ T, for all B ∈ S. An invertible topomorphism with a continuous 
inverse is a topeomorphism. 

Clearly, the projections πj: ∏ 𝐺𝑖𝑖  → Gj are topomorphisms and also the natural map q : G → G/H is also a 
topomorphism. If H is a subgroup of a topo-group (G,T), then the inclusion map j : (H,Tind) → (G,T) is a 

topomorphism, too. 

 

Remark 1.8. Let (G,T) be a topo-group. We can define a topology on G with the basis T. Let us denote this topology 

by T∗. A subset X ⊆ G is open if X is a union of elements of T. This topology is never Hausdorff but G is compact if 
and only if G is topo-compact. For a subgroup H ≤ G with the induced topo-system Tind, we have (T

∗)ind ⊆ (Tind)∗. 
We close this section with two more examples of interesting topo-systems on arbitrary groups. 

 

Example 1.9. Let G be a group and H ≤ K ≤ G. We define 

 

T
H,K 

= {A ≤ G : [A,K] ⊆ H} ∪  {G},                                                     (3) 

where [A,K] is the commutator subgroup generated by all commutators [a,b] = aba
−1

b
−1

, with a ∈ A and b ∈ K. 
Clearly, 1 and G belong to T

H,K
. This set is closed under intersection, so let {Ai}i∈I be a family of elements of T

H,K
. 

Suppose a1,...,an are elements from the union of Ai’s and x ∈ K. 
We have 

.                                      (4) 

Now, for any u ∈ K, we have aiua
−

i 
1
u

−1 
∈ H, so for some hn ∈ H, 

. 

Continuing this way, for some hn−1 ∈ H, we have 

.                                 (5) 

Finally, we get 

[a1a2 ...an,x] = h1h2 ...hn,                                                           (6) 

for some h1,...,hn ∈ H. This shows that ∏ 𝐴𝑖𝑖  ∈ T
H,K

. 

 

Example 1.10. Let G be group and H ≤ G be a fixed subgroup. Define 

𝑇𝑐𝑜𝑛𝑗 (H) = {A ≤ G : H ⊆ N(A)}, 

where N(A) is the normalizer of A in G. It can be verified that Tconj(H) is a topo-system on G. 

 

 

Hausdorff topo-groups 

 

Suppose (G,T) is a topo-group and for any x,y ∈ G with <x> ∩ <y> = 1, there exist A,B ∈ T with the properties 
x ∈ A, y ∈ B, and A ∩ B = 1. 

Then we say that (G,T) is a Hausdorff topo-system. A subgroup A ≤ G is weak T-closed, if for all x with <x> ∩ A = 1, 

there exists B ∈ T such that A ∩ B = 1. Recall that, in each ordinary Hausdorff topological space, every compact 
subset is closed. By a similar argument as in the ordinary topological spaces, one can see that the following 

proposition holds. 



TOPO-GROUPS AND A TYCHONOFF TYPE THEOREM  

103 

 

 

Proposition 2.1. Let (G,T) be a Hausdorff topo-group and A be a topo-compact subgroup. Then A is weak T-closed. 

 

Clearly, any discrete topo-group is Hausdorff. If G is infinite and (G,Tcf) is Hausdorff, then for all non-identity 

x,y ∈ G, we have 
< 𝑥 >∩< 𝑦 >≠ 1                                                                            (7) 

 

so the intersection of any two non-trivial subgroups of G is non-trivial. Clearly, such a group is torsion free or a p-

group for some prime. The groups Z, Q and Zp∞ are examples of such groups. Torsion free non-abelian groups with 
this property are constructed by Adian and Olshanskii, [5]. An elementary argument shows that G can be embedded 

in Q if G is an R-group (i.e., a group satisfying x
m 

= y
m 

⇒ x = y for any non-zero m). The next proposition shows that 
the p-group case can be investigated by a very elementary argument, and so the hardest part of the problem is the case 

of torsion free groups which are not R-groups. 

 

Proposition 2.2. Let G be an infinite p-group which is Hausdorff with respect to the topo-system Tcf . Then G has a 

unique subgroup of order p. The converse is also true. 

Proof. Let G be a p-group and the intersection of any two non-trivial subgroups of G be non-trivial. Suppose x is a 

non-identity element of G. Then <x> has a subgroup A of order p. For any non-identity subgroup B ≤ G, the 

intersection A∩B is non-trivial, so A ⊆ B, and hence A is a unique subgroup of G of order p. Conversely, suppose G 
has a unique subgroup of order p, say A. Then clearly, for any non-trivial subgroup B, we have A ⊆ B, and hence 
every two non-trivial subgroups of G have a non-trivial intersection.  

 

Note that the subgroup A in the proof of the above proposition is minimum in the set of non-trivial normal 

subgroups of G. Therefore, by a well-known theorem of Birkhoff (see [1]), such a group G is sub-directly irreducible. 

 

Corollary 2.3. Any infinite p-group which is Hausdorff with respect to the topo-system Tcf is sub-directly irreducible. 

 

For any group G, the normal topo-group (G,Tn) is Hausdorff if and only if G satisfies 

<x> ∩ <y> = 1 ⇒ <x
G
> ∩ <y

G
> = 1,                                                             (8) 

where <x
G
> and <y

G
> denote the normal closure of <x> and <y>, respectively. In the next theorem, we give some 

properties of this kind of groups. Note that a Q-group is a group in which every element x has a unique m-th root x
1/m

, 

for every non-zero integer m. 

 

Theorem 2.4. Let G be a torsion free group which is Hausdorff with respect to the topo-system Tn. Then for any x,u ∈ 
G, there are non-zero integers m and k such that ux

k
u

−1 
= x

m
. Further, if G is also a Q-group, then there exists a 

family of normal subgroups Ai, such that 

1- 𝐺 = ⋃ 𝐴𝑖𝑖 , 
2- 𝐴𝑖 ≅ 𝑄, 
3- 𝐴𝑖 ∩ 𝐴𝑗 = 1. 

As a result, G is abelian. 

Proof. Let A = G \ 1 and define a binary relation on A by 

x ≡ y ⇔ <x> ∩ <y> ≠ 1. 
This is an equivalence relation on A. Let E(x) be the equivalence class of x and {Ei = E(xi) : i ∈ I} the set of all such 
classes. Then we have 

𝐺 = ⋃ ⋃ < 𝑦

𝑦∈𝐸𝑖𝑖

> 

Now, since 

<x> ∩ <y> = 1 ⇒ <x
G
> ∩ <y> = 1,                                                              (9) 

so,  

< 𝑥 𝐺 >⊆ ⋃ < 𝑦

𝑦∈𝐸(𝑥)

>. 

 

Let u ∈ G be arbitrary. Then there is y ∈ E(x) such that uxu
−1 

∈ <y>. Hence, for some i, m and k we have 
uxu

−1 
= y

i
,  x

m 
= y

k
. 

Therefore, for any x,u ∈ G, there are non-zero integers m and k such that ux
k
u

−1 
= x

m
. Now, suppose G is a Q-group. 

Then, we have 

Ei = {y ∈ G : ∃m,k (non-zero)  x
m

 = y
k
} 

 ⊆ {𝑥𝑖
α

 : α ∈ Q} 

 ⊆ G. 



MOHAMMAD SHAHRYARI
 

104 

 

Suppose Ai = {𝑥𝑖
α

 : α ∈ Q}. Then, any Ai is a subgroup of G and clearly, we have 1-2-3. Note that, since for any u ∈ 
G we have uxiu

−1 
= 𝑥𝑖

α, for some α ∈ Q, so Ai is a normal subgroup. Now, [Ai, Aj] ⊆ Ai ∩ Aj, so, by 3, we have [Ai, Aj] 
= 1. This shows that G is abelian.  

 

We can use the Malcev completion of torsion free locally nilpotent groups (see [6] and [7]) to prove the next result 

on Hausdorff groups with respect to Tn. 

 

Theorem 2.5. Let G be a torsion free locally nilpotent group which is Hausdorff with respect to Tn. Then G is 

abelian. 

Proof. The same argument as above shows that for any x,u ∈ G, there are non-zero integers m and k such that ux
k
u

−1 
= 

x
m
. Now, suppose G∗

 
is the Malcev completion of G. We know that G∗

 
is a Q-group. By the notations of the above 

proof, let Ai = {𝑥𝑖
α

 : α ∈ Q}, which is a subgroup of G∗. A similar argument shows that 
1 −  𝐺 = ⋃ 𝐴𝑖𝑖 , 

     2 −  𝐴𝑖 ≅ 𝑄, 
3 −  𝐴𝑖 ∩ 𝐴𝑗 = 1. 

Since xjxix
−

j 
1 

= 𝑥𝑖
α

 , for some α ∈ Q, so [xi, xj] ∈ Ai and similarly it also belongs to Aj. This shows that xi and xj are 
commuting. Therefore, we have [x

p
i ,x

q
j] = 1 for all p,q∈ Q . So, [Ai ,Aj] = 1, which shows that G is abelian.  

 

Filters of Subgroups 

For standard notions of filter theory, the reader could use [8]. A filter of subgroups (or a subgroup filter) is any 

subset F ⊆ Sub∗(G), with the following properties: 

a) G ∈ F. 
b) If A ≤ B ≤ G and A ∈ F, then B ∈ F. 
c) If A and B ∈ F, then A ∩ B ∈ F. 

 

Example 3.1. For any group G, the sets {G} and Sub∗(G) are trivial examples of subgroup filters. For each x ∈ G, 
there exists the principal filter of subgroups 

Fx = {A ∈ Sub∗(G) : x ∈ A}. 
If G is an infinite group, then the set 

Fcf = {A ≤ G : [G : A] < ∞} 
is the co-finite filter of subgroups. 

Like ordinary filters, any filter of subgroups has finite intersection property (”fip” for short): 

𝐴1, … , 𝐴𝑛 ∈ 𝐹 ⇒ ⋂ 𝐴𝑖

𝑛

𝑖=1

≠ 1. 

On the other hand, a subset S ⊆ Sub∗(G) with fip is contained in a unique minimal filter of subgroups (which is the 
subgroup filter generated by S); if we define 

, 

then the subgroup filter generated by S is equal to 

FS = {B ≤ G : A ⊆ B for some A ∈ S∗}. 
 

Let P(G) be the power set of G and F1 ⊆ P(G) be an ordinary filter. Then clearly the set F = F1 ∩Sub∗(G) is a filter 
of subgroups. The converse is also true: 

 

Proposition 3.2. Let F ⊆ Sub∗(G) be a subgroup filter. Then  there is an ordinary filter F1 ⊆ P(G) such that 
F = F1 ∩ Sub∗(G). 

Proof. Let 

 

F1 = {X ⊆ G : A ⊆ X for some A ∈ F}. 
It can be easily verified that F1 is an ordinary filter and F = F1 ∩ Sub∗(G). 

  

Definition 3.3. An ultra-filter of subgroups is a subgroup filter F ⊆ Sub∗(G) such that for any finite set of subgroups 
A1,...,An, the condition ∪ iAi ∈ F implies Ai ∈ F, for some i. 
 

Proposition 3.4. Let F1 ⊆ P(G) be an ordinary ultra-filter. Then F = F1 ∩ Sub∗(G) is an ultra-filter of subgroups. 
Proof. Suppose A1 ∪ A2 ∪ ···∪ An ∈ F, where A1, A2, ..., An are subgroups. Suppose by contrary that none of Ais belong 
to F. Then, clearly, none of Ais belong to F1, and as F1 is an ordinary ultra-filter, we have G \ A1, G \ A2, ..., G \ An ∈ F1. 
This means that 



TOPO-GROUPS AND A TYCHONOFF TYPE THEOREM  

105 

 

⋂(𝐺 ∖ 𝐴𝑖)

𝑛

𝑖=1

∈ 𝐹1 

and hence, 

                                                                    ∅ = (⋂ (𝐺 ∖ 𝐴𝑖 )
𝑛
𝑖=1 ) ∩ (⋃ 𝐴𝑖

𝑛
𝑖=1 ) ∈ 𝐹1                                                               (10) 

which is a contradiction. 

  

Proposition 3.5. Let F be a subgroup filter on G. Then there exists an ultra-filter of subgroups containing F. 
Proof. We have F = F1 ∩ Sub∗(G) for some ordinary filter F1, but there exists an ordinary ultra-filter F2 such that F1 ⊆ 
F2. Now, F* = F2 ∩Sub∗(G) is a subgroup ultra-filter containing F.  
 

Definition 3.6. Let G and H be two groups and F ⊆ Sub∗(G) be a filter of subgroups. Let f : G → H be a group 
homomorphism and define 

f∗(F) = {A ≤ H : f
−1

(A) ∈ F}.                                                               (11) 
One can see that this is a subgroup filter on H and further, it is a subgroup ultra-filter, if F is so. 

The following definition connects two notions of topo-groups and ultrafilters of subgroups. 
 

Definition 3.7. Let (G,T) be a topo-group and F ⊆ Sub∗(G) be a subgroup ultra-filter. Let y ∈ G. We say that F 

converges to y, if for all A ∈ T with y ∈ A, we have A ∈ F. In this situation we write F → y. 

Note that if (G,T) and (H,S) are topo-groups, and if f : G → H is a topomorphism, then for an ultra-filter of 

subgroup F, the condition F → x implies f∗(F) → f(x), for all x ∈ G. We are now ready to prove the main theorem of 

this section: 

Theorem 3.8. Let (G,T) be a topo-group. Then G is topo-compact if and only if any ultra-filter of subgroups 

converges to some point in G. 

Proof. Let G be topo-compact and F be an ultra-filter of subgroups on G. Suppose by contrary that F does not 

converge to any element of G. So, for all y ∈ G, there exists a T-open subgroup Ay ≤ G such that y ∈ Ay and Ay does 

not belong to F. We have G = ∪ yAy, so G can be covered by a finite number of Ay’s, say 𝐴𝑦1 , … , 𝐴𝑦𝑛 . Hence 

𝐴𝑦1 ∪ ⋯ ∪ 𝐴𝑦𝑛 = 𝐺 ∈ 𝐹 

and therefore 𝐴𝑦𝑖 ∈ 𝐹, for some i, since F is an ultra-filter. This is a contradiction, so F has at least one convergence 

point in G. 

Now, suppose that G is not topo-compact. Hence, there is a covering G = ∪𝑖∈𝑗 𝐴𝑖 , where each Ai is T-open and 

𝐺 ≠∪𝑖∈𝑗 𝐴𝑖  for any finite subset J ⊆ I. This means that the set 

S = {G \ Ai   : i ∈ I}                                                                     (12) 

has ‘set-fip’ (i.e., the intersection of any finite number of its elements is non-empty) and so there exists an ordinary 

ultra-filter F1 containing S. Let F = F1 ∩ Sub∗(G). Then F is a subgroup ultra-filter. For any y ∈ G, we have y ∈ Ai for 

some i. But as F1 is an ultra-filter, Ai does not belong to F1 and already it does not belong to F. This shows that F does 

not converge to y.  

We say that two elements x and y in a group G are cyclically distinct if <x> ∩ <y> = 1. So, the above theorem says 

that a topo-system is topo-compact if and only if every ultra-filter of subgroups has at least one point of convergence 

up to the cyclic distinction. A similar assertion holds for Hausdorff topo-groups. 

Theorem 3.9. A topo-group (G,T) is Hausdorff if and only if any ultra-filter of subgroups in G converges to at most 

one point (up to cyclic distinction). 



MOHAMMAD SHAHRYARI
 

106 

 

Proof. Let F be an ultra-filter of subgroups in G and (G,T) be Hausdorff. Let x and y be cyclically distinct but F → x 

and in the same time F → y. We know that there are A, B ∈ T containing x and y, respectively and A ∩ B = 1. But then 

we have also A, B ∈ F and so 1 = A ∩ B ∈ F, which is impossible. 

Conversely, let (G,T) be not Hausdorff. Then there are cyclically distinct x and y, such that for all T-open 

subgroups A and B with x ∈ A and y ∈ B, we have 𝐴 ∩ 𝐵 ≠ 1. Suppose S = Fx ∪  Fy. Then clearly S has fip and hence 

there exists an ultra-filter of subgroups, say F, such that S ⊆ F. Now suppose A ∈ T and x ∈ A. Then 𝐴 ∈ 𝑆 ⊆ 𝐹 and 

therefore F → x. Similarly, we have F → y.  

A Tychonoff Type Theorem 

It seems that many known theorems of topology have versions in topo-groups. Once we prove such a theorem, it is 

possible to translate it into the language of groups and find an interesting theorem of group theory. In this section, we 

prove an analogue of the compactness theorem of Tychonoff (see [8]) for topo-groups. 

Theorem 4.1. Let {(Gi,Ti)}i∈I be a family of topo-compact topo-groups. Then G = ∏ 𝐺𝑖𝑖  is also topo-compact. 

Proof. Let F ⊆ Sub∗(G) be a subgroup ultra-filter. We prove that it converges to at least one point in G. Note that 

(πi)∗(F) is an ultra-filter of subgroups in Gi and since Gi is topo-compact, so it converges to some point xi ∈ Gi. 

Suppose x = (xi) ∈ G. We prove that F → x. Let A ≤ G be T-open and x ∈ A. We may assume that 𝐴 = ∏ 𝐴𝑖𝑖 , with Ai ∈ 

Ti and such that almost all Ai = Gi. Since xi ∈ Ai, so Ai ∈ (πi)∗(F), i.e. πi
−1

(Ai) ∈ F. Now, clearly 

𝐴 = ⋂ π𝑖
−1(𝐴𝑖 )

𝑖

, 

and the intersection consists of finitely many non-trivial elements of F, so it belongs to F and this shows that F → x, 
therefore G is topo-compact.  

Conflict of interest 

The author declares no conflict of interest. 

Acknowledgment 

 The author would like to thank M.H. Jafari for the useful discussion and I. Al-Ayyub for Arabic translation of 

the abstract.  

References 

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3. Passman, D.S.  The algebraic structure of group rings, Wiley, New York, London, Sydney, 1977. 

4. Thomas, J. Set theory, Springer, 2006, 73-90, Berlin, Heidelberg. 
5. Olshanskii, A.Y.  Geometry of defining relations in groups, Kluwer Academic Publishers, 1991, 296-307. 

6. Majewicz, S. On classes of exponential A-groups, Comm. Alg., 2010, 38(4), 1363-1384. 
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Géométrie, 1994, 350(1), 75-84.  

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Received   22 September 2021    

Accepted   26 April 2022