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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 Al-Manarah, Vol. 12, No. 2, 2006 . 9 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] . Al-Manarah, Vol. 12, No. 2, 2006 . 10 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. Al-Manarah, Vol. 12, No. 2, 2006 . 11 Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora 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 . Al-Manarah, Vol. 12, No. 2, 2006 . 12 Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora 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 Al-Manarah, Vol. 12, No. 2, 2006 . 13 Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora 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. Al-Manarah, Vol. 12, No. 2, 2006 . 14 Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora (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 qx 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. Al-Manarah, Vol. 12, No. 2, 2006 . 15 Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora 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. Al-Manarah, Vol. 12, No. 2, 2006 . 16 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 ……. V1 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]) nZ . Consider the topological space (X, (i.e. is considered as a subspace of (R, l.r )). Al-Manarah, Vol. 12, No. 2, 2006 . 17 l.r ) Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora Let V2 n1 (,2n 1) X , n Z . Then V2 n1 is a minimal open set at 2n+1. Let V2 n1 : 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: Al-Manarah, Vol. 12, No. 2, 2006 . 18 Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora 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 ) x0 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. Al-Manarah, Vol. 12, No. 2, 2006 . 19 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[ x0 ] 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: Al-Manarah, Vol. 12, No. 2, 2006 . 20 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: Al-Manarah, Vol. 12, No. 2, 2006 . 21 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. Al-Manarah, Vol. 12, No. 2, 2006 . 22 Minimal Open Set and Homogeneous Spaces …..…………………………..… Ali Fora 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 Al-Manarah, Vol. 12, No. 2, 2006 . 23