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The greatest splitting topology and semiregularity D. N. Georgiou and S. D. Iliadis University of Patras, Department of Mathematics, 265 04 Patras, Greece. E-mail: [email protected] E-mail:[email protected] 1 It is well known that (see, for example, [1], [2], and [3]) the intersection of all admissible topologies on the set C (Y; Z ) of all continuous maps of an arbitrary space Y into an arbitrary space Z , is always the greatest splitting topology. However, this intersection maybe not admissible. [1] H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Transactions of the American Mathematical Society, Vol. 336, No. 1 (1993), 101-119. [2] M. Escardo, J. Lawson, and A. Simpson, Com- paring cartesian closed categories of (core) compactly generated spaces, Topology and its Applications 143(2004), 105-145. [3] D.N. Georgiou, S.D. Iliadis, and F. Mynard, Function Space Topologies. Open Problems in Topology 2, Elsevier, Edited By Elliott Pearl. 2 In the case, where Y is a locally compact Hausdor space the compact-open topology on the set C (Y; Z ) is splitting and admissible (see [4], [5], and [6]), which means that the intersection of all admissible topologies on C (Y; Z ) is admissible. [4] R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51(1945), 429-432. [5] R. Arens, A topology for spaces of transformations, Ann. of Math. 47(1946), 480-495. [6] R. Arens and J. Dugundji, Topologies for function spaces, Pacic J. Math. 1(1951), 5-31. 3 In [6] an example of a non-locally compact Hausdor space Y is given having the same property for the case, where Z = [0; 1], that is on the set C (Y; [0; 1]) the compactopen topology is splitting and admissible. This space Y is the set [0; 1] with a topology , whose the semi-regular reduction coincides with the usual topology on [0; 1]. Also, in [6] (Theorem 5.3) another example of a nonlocally compact space Y is given such that the compactopen topology on the set C (Y; [0; 1]) is distinct from the greatest splitting topology. 4 In this paper, rst we prove that if two spaces (Y; 0) and (Y; 1) have the same semi-regular reduction, then for any regular space Z the sets C ((Y; 0); Z ) and C ((Y; 1); Z ) coincide and a topology t on C ((Y; 0); Z ) is splitting or admissible if and only if t is respectively splitting or admissible on C ((Y; 1); Z ). Using this fact, we construct non-locally compact Hausdor spaces Y such that the intersection of all admissible topologies on the set C (Y; Z ), where Z is an arbitrary regular space, is admissible. Furthemore, for a Hausdor splitting topology t on C (Y; Z ) we nd suÆcient conditions in order that t to be distinct from the greatest splitting topology. Using this result, we construct some concrete non-locally compact spaces Y such that the compact-open topology on C (Y; Z ), where Z is a Hausdor space, is distinct from the greatest splitting topology. Finally, we give some open problems. 5 1 Preliminaries Let Y and Z be two spaces. If t is a topology on the set C (Y; Z ) of all continuous maps of Y into Z , then the corresponding space is denoted by Ct(Y; Z ). A topology t on C (Y; Z ) is called splitting if for every space X , the continuity of a map g : X Y ! Z implies that of the map gb : X ! Ct(Y; Z ) dened by relation gb (x)(y ) = g (x; y ) for every x 2 X and y 2 Y . A topology t on C (Y; Z ) is called admissible if for every space X , the continuity of a map f : X ! Ct(Y; Z ) implies that of the map ff : X Y ! Z dened by relation ff(x; y ) = f (x)(y ) for every (x; y ) 2 X Y (see [5], [4], and [6].) 6 Let t be a topology on C (Y; Z ). If a net ff; 2 M g of C (Y; Z ) converges topologically to an element f of t C (Y; Z ), then we write ff; 2 M g ! f . (If M is the set ! of all non-negative integers, then the net is called a sequence). A net ff; 2 M g of the set C (Y; Z ) converges continuously to f 2 C (Y; Z ) (see [7] and [8]) if and only if for every y 2 Y and for every open neighborhood W of f (y) in Z there exists an element 0 2 M and an open neighborhood V of y in Y such that for every 0, f(V ) W . [7] O. Frink, Topology in lattices, Trans. Amer. Math. Soc. 51(1942), 569-582. [8] C. Kuratowski, Sur la notion de limite topologigue d' ansembles, Ann. Soc. Pol.Math. 21(1948-49), 219225. 7 The point-open topology (see, for example, [9]) on C (Y; Z ) is the topology for which the family of all sets of the form (y; U ) = ff 2 C (Y; Z ) : f (y) 2 U g; where y 2 Y and U is an open subset of Z , compose a subbasis. [9] J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass. 1966. The compact-open topology (see [4] and [5]) on C (Y; Z ) is the topology for which the family of all sets of the form (K; U ) = ff 2 C (Y; Z ) : f (K ) U g; where K is a compact subset of Y and U is an open subset of Z , compose a subbasis. 8 Let O(Y ) be the family of all open sets of the space Y . The Scott topology on O(Y ) (see, for example, [10] and [11]) is dened as follows: a subset IH of O(Y ) is open in the Scott topology if: () U 2 IH , V 2 O(Y ), and U V imply V 2 IH , and ( ) for every collection of open sets of Y , whose union belongs to IH , there extsts a nite subcollection whose union also belongs to IH . [10] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott, A Compendium of Continuous Lattices, Springer, Berlin-Heidelberg-New York 1980. [11] B.J. Day and G. M. Kelly, On topological quotient maps preserved by pull backs or products, Proc. of the Cambridge Phil. Soc. 67(1970), 553-558. 9 The Isbell topology (see, for example, [12], [13], and [14]) on C (Y; Z ) is the topology for which the family of all sets of the form (IH; U ) = ff 2 C (Y; Z ) : f 1 (U ) 2 IH g; where IH is a Scott open set of Y and U is an open subset of Z , compose a subbasis. [12] P. Lambrinos and B.K. Papadopoulos, The (strong) Isbell topology and (weakly) continuous lattices, Continuous Lattices and Applications, Lecture Notes in pure and Appl. Math. No. 101, Marcel Dekker, New York 1984, 191-211. [13] R. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Lecture Notes in Mathematics, 1315, Springer Verlang. [14] F. Schwarz and S. Weck, Scott topology, Isbell topology, and continuous convergence, Lecture Notes in Pure and Appl. Math. No.101, Marcel Dekker, New York 1984, 251-271. 10 A subset B of a space X is called bounded (see, for example, [12]) if every open cover of X contains a nite subcover of B . A space X is called corecompact if for every open neighborhood U of a point x 2 X there exists an open neighborhood V U of x such that V is bounded in the space U (see, for example, [10], [12], and [14]). 11 It is well known that: (1) A topology t on C (Y; Z ) is splitting if and only if each net of elements of C (Y; Z ) converging continuously to an element of C (Y; Z ) converges also topologically to this element (see [6]). (2) On the set C (Y; Z ) there exists the greatest splitting topology (see [6]). (3) The intersection of all admissible topologies coincides with the greatest splitting topology (see, for example, [1], [2], and [3]). (4) The compact-open topology, denoted here by tco, is always splitting (see [4] and [5]) and, in general, does not coincide with the greatest splitting topology (see [6]). For a regular locally compact space Y the topology tco is always admissible and, therefore, coincides with the greatest splitting topology (see [4], [5], and [6]). 12 (5) The Isbell topology, denoted here by tIs, is always splitting (see, for example, [12], [13], and [14]) and, in general, does not coincide with the greatest splitting topology (see [15] and [16]). For a corecompact space Y the topology tIs is always admissible and, therefore, coincides with the greatest splitting topology. (6) If Z is a Hausdor space, then the topologies tco and tIs are Hausdor (see, [7] and [13]). [15] M. Escardo, J. Lawson, and A. Simpson, Com- paring cartesian closed categories of (core) compactly generated spaces, Topology and its Applications 143(2004), 105-145. [16] D.N. Georgiou and S.D. Iliadis, On the compact open and nest splitting topologies. Topology and its Applications, In Press. 13 A space Y is called Z -harmonic if the compact-open topology coincides with the greatest splitting topology on C (Y; Z ). If Y is Z -harmonic for every space Z , then Y is called harmonic (see [3]). A space Y is called Z -concordant if the Isbell topology coincides with the greatest splitting topology on C (Y; Z ). If Y is Z -concordant for every space Z , then Y is called concordant (see [3]). Let (Y; ) be a space. Consider the subset fInt(Cl(U )) : U 2 g of . The set b is a base for a topology on Y , denoted here by sr . The space (Y; sr ) is called semi-regular reduction of (Y; ). Obviously, sr . A topology is called semi-regular if = sr . Therefore, a topology is semi-regular if and only if there exists a base for the topology such that U = Int(Cl(U )) for every element U of this base. It is easy to verify that: () sr is a semi-regular topology and ( ) any regular topology is semi-regular. We say that two spaces (Y; 0) and (Y; 1) have the same semi-regular reduction if sr0 = sr1 . 14 2 The greatest splitting and admissible topologies In this section we study the relations between the splitting and admissible topologies on two function spaces C ((Y; 0); Z ) and C ((Y; 1); Z ) in the case, where the spaces (Y; 0) and (Y; 1) have the same semi-regular reduction. Proposition 2.1. Let (Y; 0); (Y; 1), and Z be spaces such that C ((Y; 0); Z ) = C ((Y; 1); Z ) and for every space X , C (X (Y; 0); Z ) = C (X (Y; 1); Z ): Then, a topology t on C ((Y; 0); Z ) is splitting (respectively, admissible) if and only if t is splitting (respectively, admissible) on C ((Y; 1); Z ). Proposition 2.2. Let (Y; 0), (Y; 1) be two spaces with the same semi-regular reduction and Z a regular space. Then, C ((Y; 0); Z ) = C ((Y; 1); Z ) and for every space X , C (X (Y; 0); Z ) = C (X (Y; 1); Z ): 15 (1) (2) Let (Y; 0), (Y; 1) be two spaces with the same semi-regular reduction and Z a regular space. Then, a topology t on C ((Y; 0); Z ) is splitting (respectively, admissible) if and only if t is splitting (respectively, admissible) on C ((Y; 1); Z ). Therefore, the intersection of all admissible topologies on C ((Y; 0); Z ) is admissible if and only if the intersection of all admissible topologies on C ((Y; 1); Z ) is admissible. Proposition 2.4. Let Y be a space whose the semi-regular reduction is a regular locally compact space and Z an arbitrary regular space. Then, the intersection of all admissible topologies on C (Y; Z ) is admissible and, therefore, the greatest splitting topology is admissible. Corollary 2.3. The following proposition is a generalization of Proposition 2.4. Let Y be a space whose the semi-regular reduction is corecompact and Z an arbitrary regular space. Then, the intersection of all admissible topologies on C (Y; Z ) is admissible and, therefore, the greatest splitting topology is admissible. Proposition 2.5. 16 Similarly to the above two propositions we can prove the following propositions. Let Y be a space whose the semi-regular reduction Y sr is harmonic and Z an arbitrary regular space. Then, the greatest splitting topology on C (Y; Z ) is the compact-open topology on C (Y sr ; Z ). Proposition 2.7. Let Y be a space whose the semi-regular reduction Y sr is concordant and Z an arbitrary regular space. Then, the greatest splitting topology on C (Y; Z ) is the Isbell topology on C (Y sr ; Z ). Proposition 2.6. 17 1. Let [0; 1] be the closed interval of the real line and let M = f1=n : n 2 ! g. Denote by Y the closed interval [0; 1] with the topology 1 for which the family 0 [ fY n M g; where 0 is the usual topology of [0; 1], compose a subbase. By Proposition 2.5, the greatest splitting topology on the set C (Y; [0; 1]) is admissible (see also Theorem 6.21 of [6]). 2. Let (Y; 0) be an arbitrary space and M a subset of Y with the property that Cl(Y n M ) = Y . On the set Y we consider the topology 1 for which the family 0 [fY n M g compose a subbase. The spaces (Y; 0) and (Y; 1) have the same semi-regular reduction. By Proposition 2.5, for every corecompact space (Y; 0) and an arbitrary regular space Z , the greatest splitting topology on the set C ((Y; 1); Z ) is admissible. Examples 2.8. 18 3. Let R be the set of real numbers. On the set R we consider a topology 0 for which the set f[a; 1) : a 2 Rg [ f;g compose a base and a topology 1 for which the set f(1; a) : a 2 Rg [ f;g compose a base. Note that the space (R; 0) is corecompact but not locally compact and the space (R; 1) is locally bounded but not corecompact or locally compact. It is easy to see that the semi-regular reduction of these spaces is the set R with the trivial topology. By Proposition 2.5, the greatest splitting topology on the set C ((R; 1); Z ), where Z is an arbitrary regular space, is admissible. 19 3 On the splitting topologies Let t be a splitting Hausdor topology on C (Y; Z ), where Y and Z are arbitrary spaces. If there exist a sequence ffi : i 2 ! g of elements of C (Y; Z ), an element f1 of C (Y; Z ), and a t point y0 of Y such that: (a) ffi : i 2 ! g ! f1 and (b) there exists an open neighborhood W0 of f1(y0) in Z such that for every open neighborhood U 0 of y0 in Y there exists i(U 0 ) 2 ! for which fi (U 0) 6 W0 for every i i(U 0 ) (therefore, t is not an admissible topology), then t is not the greatest splitting topology on C (Y; Z ). Corollary 3.2. Let Y be an arbitrary space. If there exists a Hausdor space Z such that the compactopen topology t = tco on C (Y; Z ) satises the conditions of the Proposition 3.1, then Y is not harmonic. Corollary 3.3. Let Y be an arbitrary space. If there exists a Hausdor space Z such that the Isbell topology t = tIs on C (Y; Z ) satises the conditions of the Proposition 3.1, then Y is not concordant. Proposition 3.1. 20 Let Yi, i 2 ! , be a family of Hausdor spaces such that Yi \ Yj = ; if i 6= j . Suppose that for every i 2 ! there exists a lter Fi of non-empty open sets of Yi with the property that Yi n K 2 Fi for every compact subset K of Yi. On the set Example 3.4. [ Y = ( fYi : i 2 !g) [ f1g; where 1 is a symbol, we consider a topology for which a subset V of Y is open if and only if: () V \ Yi is open in Yi for all i 2 ! and ( ) in the case, where 1 2 V , there exists a nite subset s of ! such that V \ Yi 2 Fi for all i 2 ! n s. We note that the space Y has the properties: (1) Yi is simultaneously open and closed subspace of Y . (2) If K is a compact subset of Y , then there exists a nite subset s of ! such that K [fYi : i 2 sg [ f1g. 21 Let Z be an arbitrary Hausdor space containing two distinct points a and b. Consider the sequence ffi; i 2 ! g of maps of Y into Z for which fi (y ) = b if y 2 Yi and fi(y) = a if y 2 Y n Yi. Using the above properties (1) and (2) one can prove that the the maps fi are elements of C (Y; Z ) and the sequence ffi ; i 2 ! g converges to f1 in the compact-open topology, where f1 is the element of C (Y; Z ) dened by condition f1 (y ) = a for all y 2 Y , which means that the condition (a) of the Theorem 2.1 is satised. We prove that condition (b) of this theorem is also satised. Indeed, let W0 be an open neighborhood of a which does not contain the point b. Consider an arbitrary open neighborhood U 0 of 1 in Y . Then, there exists a nite subset s of ! such that U 0 \ Yi 6= ; for every i 2 ! n s. Setting i(U 0 ) = maxfi : i 2 sg we have that fi (U 0 ) 6 W0 for every i i(U 0 ) proving condition (b). Since Z is a Hausdor space the compact-open topology tco on C (Y; Z ) is a splitting Hausdor topology and, therefore, by Proposition 3.1, tco is not the greatest splitting topology on C (Y; Z ). By Corollary 3.2, the space Y is not harmonic. 22 The above example can be considered as a generalization of the space Y considered in the Theorem 5.3 of [6]. Remark 3.5. 4 Some open problems Let P be a class of spaces having the same semi-regurar reduction and Z a regular space. By Proposition 2.2 and Corollary 2.3 the set C (Y; Z ) and the greatest splitting topology on this set are independent of the elements Y of P . (1) Is the compact-open topology on C (Y; Z ) independent of the elements Y of P ? (2) Is the Isbell topology on C (Y; Z ) independent of the elements Y of P ? (3) Suppose that P contains an element which is Z harmonic (respectively, harmonic). Is any element of P Z -harmonic (respectively, harmonic)? (4) Suppose that P contains an element which is Z concordant (respectively, concordant). Is any element of P Z -concordant (respectively, concordant)? Problems 4.1. 23 It is known that the compact-open topology does not coincide with the greatest splitting topology on the set C (N ! ; N ), as well as, on the set C (R! ; R), where N is the set of natural numbers with the discrete topology and R is the set of real numbers with the usual topology. (1) Suppose that the space Z is the space N or the space R. Can be nd a sequence ffi : i 2 !g of elements of C (Z ! ; Z ), an element f1 of C (Z ! ; Z ), and a point y0 2 Z ! such that the conditions (a) and (b) of the Proposition 3.1 to be satised? (2) Let Y and Z be two spaces such that the compactopen (respectively, the Isbell) topology on C (Y; Z ) is not the greatest splitting topology. Under what (internal) conditions on Y and Z the conditions of the Proposition 3.1 (concerning the set C (Y; Z )) are satised? Problems 4.2. 24