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Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora
Minimal Open Set and Homogeneous Spaces
Received: 5/2/2003
Accepted : 2/6/2003
Ali Ahmad Fora*
‫ملخص‬
‫ لقد ناقشنا في هذا البحث‬.‫لقد صنفنا بعض الفضاءات المتجانسة الالنهائية المعدودة‬
‫المجموعات المفتوحة األصغرية محليا واستخدمنا هذا المفهوم التبولوجي في تصنيف بعض‬
.‫ وأخي ار تم عرض بعض األمثلة والمسائل المفتوحة‬.‫الفضاءات التبولوجية‬
Abstract
We characterize some classes of infinite countable (denumerable)
homogeneous topological spaces. The concept of having minimal open
neighborhoods Will be studied and we use it in a characterization result through
the paper. Finally, we provide some examples relevant to the results we have
obtained, and we also propose some open problems.
1. Introduction
If X is a topological space, then H(X) denotes the set of all auto
homeomorphisms on X. A topological space X is called homogeneous if
for any x, y  X there is an h  H ( X ) such that h(x)=y. The concept of
homogeneity was introduced by W.Sierpinski [3]. Several mathematicians
have studied homogeneous spaces. One may consult [1] for this study .
X
Let X be any set.
denotes the cardinality of X . By .  dis ,  ind and
 cof we mean the discrete, indiscrete and the co-finite topologies on X,
respectively . N, Nk, Z, Q and R will denote the sets of all natural numbers,
{1, 2,….,k}, integers, rationals and real numbers, respectively ( k  N ).  u
and  l.r will denote the usual(Euclidean) and the left ray topologies on the

nonempty subset X  R (i.e. l.r = {  ,X,(-  ,a)  X: a  R}). If (X,  ) is
a topological space and A is a subset of X, then the closure of the set A in
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Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora
*
Assistant Professor, Department of Mathematics, Yarmouk University, Jordan.

X will be denoted by A and ,  A will denote the relative topology on A.


If A= {x} then X will stand for A .
A set X is called denumerable if it is equipotent with N. However, a set
X is called countable if it is either finite or denumerable .
A space X is said to have the fixed point property if and only if every
continuous map f : X  X has a fixed point ( i.e. a point x in X such that
f(x) = x ).
Several authors studied finite topological spaces with some topological
Structures. For example , Fora and Al-Bsoul [2] gave the following
characterization of certain homogenous spaces .
THEOREM 1.1.
Let (x,  ) be a topological space which contains a nonempty open
indiscrete subset (n the induced topology). Then the following are
equivalent:
a) (X,  ) is a homogeneous space.
b) (X,  ) is a disjoint union of indiscrete topological space all of which
are homeomorphic to one another .
Then they (Fora and Al-Bsoul) used Theorem1.1 to obtain the
following characterization result.
THEOREM 1.2.
 0
Let (X,  ) be a topological space with
<
, then (X,  ) is
homogeneous if and only if (X,  ) is the sum of mutually
homeomorphic indiscrete spaces .
The following result is also obtained in [2] .
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Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora
THEOREM 1.3.
Every nondegenerate (i.e.
have the fixed point property .
X
>1) homogeneous finite space does not
2. Minimal Open Sets .
We start this section with the following definition.
DEFINITION 2.1.
Let (X,  ) be a topological space and H an open set in X containing
the point p  X. The set H is called a minimal open set at p if there is no
open proper subset of H containing p , i.e. if V is an open set in X
containing p then H  V. A nonempty open set U is called minimal if it is
minimal at each x  U.
In the topological space (Z,  l.r ), notice that the open set (-  ,9) I Z
isminimal at 8 , but not minimal at 4.
The following result is easy to observe.
THEOREM 2.2
Let (X,  ) be a topological space . Let H be a minimal open set in X at
p and let a  H. Then the following are equivalent :
i) H is not minimal at q .
ii) there exists an open set V containing q but not p .

iii) q  p .
proof. (i)  (iii) :Since the open set H is not minimal at q, therefore
there exists a proper open set V  H such that q  V. But H is minimal at
p. Therefore p  V.
(ii)  (iii) : The proof is trivial .

(iii)  (i) : If q  p then there is an open set V such that q  V and p  V.
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Now, V  H is an open proper subset of H and containing q. Hence H is
not minimal at q.
As a direct conclusion of Theorem 2.2, we have the following result.
COROLLARY 2.3.
Let ( X , ) be a To -space and let H be any open set in X containing
two distinct points p,q. Then H can not be minimal at p and at q
simultaneously.
The following result clarifies that minimal open sets have been indeed
used in Theorem1.1.
THEOREM 2.4
Let ( X , ) be a topological space and let A be a nonempty open set in
   ind .
X. Then A is minimal if and only if A
Proof. Let A be a nonempty minimal open set in X. If  A   ind then
there is V  A such that V   and V  A . It follows that there is an open
set H in X such that V  H  A . Since H and A are both open sets in X,
therefore V is a nonempty open set in X and it is properly contained in the
minimal set A which is absurd.
   ind , then
On the other hand, if A is a nonempty open set such that A
A will be minimal because if not , then there is a proper nonempty open
subset H of A. Hence H  A which is absurd because H   and H  A .
THEOREM 2.5.
A topological space X has a minimal open set at p in X if
and only if the intersection of all open sets in X containing p is still an
open set in X.
Proof. Trivial .
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COROLLARY 2.6.
If the topology  on X is finite. Then the space (X,  ) has minimal
open sets at each x in X.
THEOREM 2.7
Let X, Y be two topological spaces.
i) If U, V are minimal open sets at p, q in X, Y, respectively. Then U x V,
is a minimal open set at (p,q) in X x Y .
ii) If G is a minimal open set at (p,q) in XxY . Then there exit U, V
minimal open sets at p , q in X , Y, respectively, such that G= UxV.
Proof. i) Suppose H is an open set in X x Y such that (p,q)  H 
UxV.
Then there are open sets U1 , V1 in X,Y, respectively, such that (p,q) 
U1 x V1  H .
Thus p  U1 and q  V1 . Consequently, we have U1  U and V1  V.
Henceforth U1 = U and V 1 = V . This implies that H= U x V. Therefore U x
V is a minimal open set at (p,q).
ii) Since G is an open set in X x Y containing (p,q) . Therefore there exit
U, V open sets in X,Y respectively such that (p,q)  U x V  G. Since G is
minimal at (p,q) , therefore U x V = G. To prove U is minimal at p,
suppose that U1 is an open set in X such that p  U1  U . Then (p,q)
U1 xV  UxV  G , but G is minimal at (p,q), so U1 xV  UxV , i.e.
U1  U . Similarly, it can be shown that V is minimal at q.
THEOREM 2.8
If a T1 -space ( X , ) has a minimal open set at some point p in X, then
p is an isolated point of X.
Proof. Let H be a minimal open set at p. If there is a point q  H such
that p  q , then H  q will be an open set in T1 -space X and properly
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contained in H and this is absurd. Thus H  p and this completes our
proof of the theorem , The following is a direct consequence of Theorem
2.8.
COROLLARY 2.9.
The only homogenous T1 -space ( X , ) having a minimal open set at
some ( each ) point is the discrete space.
To state our next result we need the following definition.
DEFINITION 2.10.
A topological space ( X , ) is called regular at p  X if for any open

set G in X containing p, there is an open set U such that p  U  U  G .
However, a topological space ( X , ) is called 0-dimensiona at p  X if X
has a local base at p consisting of clopen (i.e. closed and open
simultaneously) sets in X.
A topological space ( X , ) is called 0-dimensional if X has a base
consisting of clopen sets in X . It is clear that ( X , ) is 0-dimensional if
and only if ( X , ) is 0-dimensional at each point in X.
Now, we can state our next result.
THEOREM 2.11.
Let ( X , ) be a topological space and p  X .
i) if X has a minimal open set at p, then X is first countable at p.
ii) if X has a minimal open set at p and X is regular at p, then X is 0dimensional at p.
Proof.(i) A local base at p may be obtained by taking only the minimal
open set at p.
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(ii) Let G be a minimal open set at p. Since X is a regular space at p, there

exists an open set U such that p  U  U  G . It follows that G=U and
consequently U  U  G . Hence U is a clopen set in X. Now G is a
local base at p, and this completes the proof of our theorem.
THEOREM 2.12.
Let ( X , ) be a homogenous topological space having a minimal open
set at some points. Then X is a first countable space .
If X is also regular at some oints then X is 0-dimensional.
Proof. Since X has a minimal open set at p in X, therefore ; by
Theorem 2.11; X is first countable at p. The fact X is a homogenous space
and first countable at p in X implies that X is first countable. Indeed, if Bp
is a countable local base at p and if q is any point in X, then there exists a
homeomorphism f:X  X such that f(p)=q. Now, the collection
Bq  f (U ) : U  B p 
is a countable local base at q.
To prove the second assertion of the theorem, let X be a regular space
at q in X and having minimal open set at p in X. By homogeneity of X, X
is regular at p.
Therefore; by Theorem 2.11; X is 0-dimensional at p. Now, for any q
 X, there exists a homeomorphism f:X  X such that f(p)=q. Since X is 0dimensional at p, there exists a local base Bp at p consisting of clopen sets
B  f (U ) : U  B p 
in X. Let q
. Then Bq is a local base at q consisting of
B   Bq
qx
clopen sets in X. Now, it is clear that the collection
is a base for
the topology on X consisting of clopen sets in X. Hence X is 0dimensional.
3. Denumerable Homogenous Spaces and Minimal Open Sets.
As an application of the local minimality of open sets, we shall present
the following classification of the left ray topology on the integers.
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THEOREM 3.1.
Let ( X , ) be a denumerable homogenous T0 – space and let  be a
collection of minimal open sets at some ( each ) points in X such that:
(i) for any x  X there exists Fx   such that Fx is a minimal open set at
x;
F ,F 
(ii) for any x,y  X ; x  y ; the minimal open sets x y
at x,y,
respectively, are comparable (i.e.
Fx  Fy
or
Fy  Fx
) and, such that.
(iii){ F   : F is between Fx and Fy} is a finite set Then ( X , ) is
homeomorphic to ( Z,  l.r ).
Proof. Using the assumption of the theorem, one can construct an
infinite two sided tower {V n : n  Z } (i.e. V n  Vn+1 for all n  Z) of
minimal open set Vn at x n ; where X = { x n : n  Z}. It can be easily
V Vj
V Vj
shown that j 1
= 1 for every j  Z . Now , letting j 1
= {xj+1 }
and define f: ( Z,  l.r )  (X,  ) by f(j) = x j .
Then f is a homeomorphism and our proof is completed .
The following examples show the necessity of the assumptions
mentioned in Theorem 3.1
EXAMPLE 3.2.


l. r )
In the product space X of (Z,
and ({0,1}, ind );
  {Vn  ((, n]  Z )
x {0,1} :n  Z} ; where V n is a minimal open ser at
each of the two point (n,0) and (n,1) (n  Z). It is clear that V n  V n+1 for
V Vj
all n  Z but j 1
= 2 for every j  Z. This is because X is not a T0 –
space. It is clear that the spaces X and ( Z,  l.r ) are not homeomorphic to
each other.
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Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora
EXAMPLE 3.3.
The topological space (N,  l.r ) satisfies all assumptions of Theorem 3.1
except the homogeneity . This; inhomogeneity ; affects the existence of
the infinite two sided tower …….  V1  V0  V1  ….. mentioned in the
proof of Theorem 3.1. Notice that the topological spaces (N,  l.r ) and (Z,
 l.r ) are not homeomorphic to each other.
EXAMPLE 3.4.

The topological space (N, cof ) satisfies all assumptions of Theorem 3.1
except the existence of minimal open sets at point in N. Notice that the

topological spaces (N, cof ) and (Z,  l.r ) are not homeomorphic to each
other.
EXAMPLE 3.5.
Let X be the product of two copies of (Z,  l.r ) . For m  Z, let V m =
((   ,m]x(   ,m] )  Z x Z and let   {Vn : n  Z } . For any x = (m.n)
 ZxZ , let j= max {m,n}. Then x  V j   and our space X satisfies all
assumptions of Theorem 3.1 expect that  does not satisfy (i). Actually , V
j may not be a minimal open set at (m,n) ; where j = max{m,n}. For
example (1,2)  V2 but V2 is a minimal open set at (2,2) and not at (1,2).
Notice that the topological spaces X and (Z,  l.r ) are not homeomorphic to
each other.
EXAMPLE 3.6.
Let
X  U (Q  [2n,2n  1])
nZ
. Consider the topological space (X,
(i.e. is considered as a subspace of (R,  l.r )).
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 l.r )
Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora
Let V2 n1  (,2n  1)  X , n  Z . Then V2 n1 is a minimal open set at


2n+1. Let   V2 n1 : n  Z . Then  does not satisfy (i) in Theorem 3.1.
Notice that X is indeed not homogenous.
EXAMPLE 3.7.
In the space ( Z , ind ) , there is only one minimal open set, namely Z
itself. One can not find the family  ; as stated in Theorem 3.1; because
( Z , ind ) is not a T -space.
0
EXAMPLE 3.8.

Consider
the
topology
defined
on
R
by
   , R, (, a), (, a] : a  R. Then the subspace (Q, Q ) is a
F  (, x]  Q
denumerable homogenous T -space. For any x  Q , let x
0


and let   Fx : x  Q then  does not satisfy (iii) as stated in Theorem
3.1. Hence, the condition (iii) that was stated in Theorem 3.1 can not
(Q, Q )
omitted because
and (Z,  l.r ) are not homeomorphic to each other.
4. Homogenous Spaces Having Minimal Open Sets.
In the following results we shall characterize homogenous topological
spaces having minimal open sets.
THEOREM 4.1.(A.C.)
let ( X , ) be a homogenous topological space having a minimal open
set A. Then ( X , ) is homeomorphic to the product space of ( A, A ) and a
discrete space (i.e. ( X , ) is the product of a discrete and an indiscrete
spaces).
Proof. The proof will be through the following three steps:
I. Define a relation  on X as follows:
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x  y if only and if there exists a minimal open set H in X containing
both x and y. We are going to prove that  is an equivalence relation on X.
Since X has a minimal open set and ( X , ) is a homogenous space.
Therefore  is a reflexive on X .  is trivially symmetry. Now, if H is a
minimal open set in X containing x and y; and V is a minimal open set in X
containing y and Z. Then H  V is an open set contained in both minimal
open sets H and V. Hence H= H  V =V. Consequently, we get H is a
minimal open set containing x and z. This ends the proof that  is an
equivalence relation on X.
II. Let X/  be the collection of all equivalence classes of X endowed with
the quotient topology induced by the canonical mapping j : X  X / 
defined by j ( x)  [ x], x  X . We are going to prove that X/  has indeed
the discrete topology. To prove this, let x  X . In order to prove that {[x]}
1
is open in X/  it suffices to show that j ([ x]) is an open set in X. To do
1
this , let y  j ([ x]) . Then j( y)  [ y]  [ x] , i.e.[y] = [x]. Hence there
exists a minimal open set U in X containing x and y. If z  U then U is a
minimal open set in X containing both z and y. Hence z  y . This yields
1
that U  [ y ]. Therefore y  U  [ y ]  [ x]. Thus j ([ x]) is an open set
in X and hence [x] is open in X/  . As we have seen before U=[y]. This
means that equivalence classes of X are precisely minimal open sets in X.
Now, since X/  is a partition on X, so by axiom of choice (A.C.), X/ 
P  [ x]  1
has a choice set P(i.e. is a subset of X such that
for all x in X).
  X /
P  x a :   
   such that A= [ x 0 ] .
Write
, where
. Let 0
By the homogeneity of X, for any    there exists a homeomorphism
h ( x )  x0
h : X  X
such that
. It is not difficult to notice that
h ([ x ])  A.
 ( x)  (h ( x), [ x]) ,
III. Now, define a mapping  : X  Ax( X /  ) by
where    is uniquely determined by the membership relation x  [ x ] .
We are going to prove that  is a homeomorphism.
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Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora
To prove  is an injective mapping, let  ( x)   ( y ) . Then [x] = [y]
h ( x)  h ( y )
y  [ x ]  ,   
and 
, where x  [ x ] and
;
. It follows that
[ x ]  [ x]  [ y ]  [ x  ]
x  x 
. Thus
and hence    . Since
h ( x)  h ( y) therefore x = y. Hence  is an injective mapping.
To prove  is a surjective mapping, let (t , [ x])  Ax( X /  ) , where
x  X . Then there is a unique    such that x  [ x ] . Since
h : X  X is a surjective mapping, there exists y  X such that
t  h ( y) . Notice that h ([ y])  [h ( y)]  [t ]  A because t  A . The fact
that h ([ x ])  A  h ([ y]) implies that [ x ]  [ y ] , i.e. y  [ x ]. Now, it
is clear that  ( y)  (h ( y), [ y])  (t , [ y])  (t , [ x ])  (t , [ x]) . This ends the
proof that  is a bijection.
To prove  is an open mapping it suffices to show that  (A) is open
in Ax(X/  ) because of the use of Theorem 1.1 and 2.4 since A is a
minimal open set in the homogenous space X. Observe the fact that
 ( A)  Ax[ x0 ]
will end the proof that  is an open mapping .
To prove  is a continuous mapping we need the following claim: For
any x  X
h
We have  -1 (Ax {[x]}) =  -1(A); where    is uniquely
determined by the membership relation x  [ x ]. If our claim is true then
 will be continuous because Ax{[x]} is a basic open set in Ax (X/  ).
h
To prove our claim let y  -1 (Ax{[x]}). Then  (y) =  (y),[y])
 Ax{[x]}, where y  [ x ]. Thus h (y)  A and hence y  h -1(A). To
h
h
h
prove the other inclusion, let t   -1(A). Then  (t)  A, where  :X
 X is a homeomorphism satisfying the condition that h ( x ) = x 0 . Let
 (t) = ( h (t),[t] ), where t  [ x  ] . Then [t] = [ x  ]. The following
equalities are easy to observe:
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Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora
[ h ( x ) ] = [
x 0
] = A = [ h  (t) ] = h  ( [t] ) = h  ( [
x
] ) = [h  (
x
)].
x
Therefore  =  and hence we have x =  , [t] = [ x ] = [x] and
 (t )  Ax{[ x]} i.e. t   1 ( Ax{[ x]}) .
Since  is a homeomorphism, therefore X is homeomorphic to the
product Space of the indiscrete space ( A, A ) (according to Theorem 2.4)
and the discrete space X/  .
As a direct application of Theorem 4.1 and Theorem 2.7 we have the
following result .
COLOLARY 4.2.
Let (X,  ) be a topological space having a minimal open set. Then
(X,  ) is homogeneous if and only if it is the product of two topological
spaces : one is discrete and the other is indiscrete.
Proof. If (X,  ) is a homogeneous topological space having a minimal
open set , then according to Theorem 4.1, (X,  ) is the product of a
discrete and an indiscrete spaces . Conversely, let (X,  ) be the product of
two topological spaces:
one is X1 discrete and the other X2 is indiscrete . Then X contains a
nonempty open indiscrete subset, namely { x1 } x X2 (x1  X1 ) . Since X
is the disjoint union of the indiscrete topological spaces { { x1 }x X2 : x1 
X1} all of which are homeomorphic to one another, therefore (X,  ) is a
homogeneous space according to Theorem 1.1.
Theorem 4.1, Theorem 3.1 and Theorem 2.1 suggest the following
Classification problem.
PROBLEM 4.3.
Every countable homogeneous space, (X,  ) is homeomorphic to a
finite product of some of the following spaces:
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Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora

( N k ,  dis ) , ( N k ,  ind ) , (N,  dis ) , ( N,  ind ) , ( N, cof ), (Z,  l.r ) , (Q ,  u ) ;
( k  N) (i.e. if X is a countable homogeneous space then X is

homeomorphic to a product space   Xa ; where  is a finite set and Xa is
one of the above seven classes of topological spaces).
It is worth noticing that our Problem 4.3 is true in the following three
cases:
(1) X is finite (2) (X,  ) has a minimal open set, and (3) (X,  ) satisfies
the assumption of Theorem 3.1 concerning the existence of local minimal
open sets. Theorem 1.3 and the preceding Problem 4.3 suggest the
following open problem.
PROBLEM 4.4
X
If (X,  ) is a countable homogeneous space such that
>1. Then
countable homogeneous space, does not have the fixed point property.
Problem 4.3 also suggests the following open problem.
PROBLEM 4.5.
Every countable homogenous space is first countable.
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REFERENCES
[1] B. Fitzpatrick and H.X.Zhou, A survey of some homogeneity
properties in topology, Ann. N .Y. Acad. Sci. 552(1989), 28-35.
[2] A. Fora and A. Al-Bsoul, Finite homogenous spaces, Rocky Mountain
J. of Math., vol.27, no.4 (1997), 1089-1094.
[3] W. Sierpinski, Su rune propriete topologique des ensembles
denombrables dense en soi, Fund. Math. 1 (1920), 11-28
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