gutevagt.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 5, No. 1, 2004 pp. 71- 78 Some problems on selections for hyperspace topologies Valentin Gutev and Tsugunori Nogura Abstract. The theory of hyperspaces has attracted the attention of many mathematicians who have found a large variety of its applications during the last decades. The theory has taken also its natural course and has yielded lots of problems which, besides their independent inner beauty, provide ties with numerous classical fields of mathematics. In the present note we are concerned with some open problems about selections for hyperspace topologies which have been in the scope of our recent research interests. 2000 AMS Classification: 54B20, 54C65, 54F05, 54E50 Keywords: Selections, hyperspaces, ordered spaces, complete metric spaces. 1. The concept of a τ-continuous selection Let (X, T ) be a T1-space, where T is the topology of X, and let F(X, T ) be the set of all non-empty closed (with respect to T ) subsets of X. Let us stress the reader’s attention that F(X, T ) is different for different topologies T on X, while X is always a subset of F(X, T ) because we may identify each point x ∈ X with the corresponding singleton {x} ∈ F(X, T ). Definition 1.1. A topology τ on F(X, T ) is called admissible (see [18]) if its restriction on the set of all singletons {{x} : x ∈ X} of X coincides with the topology T . In the light of Definition 1.1, we may look at (F(X, T ), τ) as a topological exten- sion of the topological space (X, T ) provided τ is an admissible topology. It should be mentioned that the concept of an admissible topology may refer also to some additional structures on X, see [18]. The second basic concept of this paper is related to a selection for a hyperspace topology. Let D ⊂ F(X, T ). Definition 1.2. A map f : D → X is a selection for D if f(S) ∈ S for every S ∈ D. Definition 1.3. If τ is a topology on F(X, T ), then a map f : D → X is a τ- continuous selection for D if it is a selection for D which is continuous with respect to the relative topology on D as a subspace of (F(X, T ), τ). So far, one of the best known admissible topologies on F(X, T ) is the Vietoris one τV (T ). Let us recall that all collections of the form 〈V〉 = { S ∈ F(X, T ) : S ⊂ ⋃ V and S ∩ V 6= ∅, whenever V ∈ V } , 72 Valentin Gutev and Tsugunori Nogura where V runs over the finite subsets of T , provide a base for the topology τV (T ). Any selection has the following property with respect to the Vietoris topology, it appeared in several papers in an explicit or implicit way. Proposition 1.4. If f : F(X, T ) → X is a selection for F(X, T ), then f is τV (T )- continuous at {x} for every x ∈ X. 2. Selections and orderability In what follows, all spaces are assumed to be at least Hausdorff. For a space (X, T ) and 0 < n < ω, we let Fn(X) = {S ⊂ X : 0 < |S| ≤ n}. Note that F1(X) is the set of all singletons of X, and always Fn(X) ⊂ F(X, T ). Let τ be a topology on F(X, T ), and let Seℓτ(X, T ) be the set of all τ-continuous selections for F(X, T ). Also, let Seℓ(τ,n)(X, T ), n > 1, be the set of all τ- continuous selections for Fn(X), and Seℓn(X) that of all selections (not necessarily τ-continuous) for Fn(X). Any selection f ∈ Seℓ2(X) naturally defines an order-like relation ≺f on X [18] by letting for x 6= y that x ≺f y iff f({x, y}) = x. However, in general, ≺f fails to be a linear order on X. Let us denote by Tf the topology generated by all possible “open” ≺f -intervals. It is easy to observe that Tf is also a Hausdorff topology, see [16]. Theorem 2.1 ([18]). Let (X, T ) be a space, and let f ∈ Seℓ(τV (T ),2)(X, T ). Then, (a) Tf ⊂ T . If, in addition, (X, T ) is connected, then we also have that (b) “≺f” is a proper linear order on X, (c) (X, Tf ) is connected, (d) f ∈ Seℓ(τV (Tf ),2) (X, Tf ). Theorem 2.2 ([20]). Let (X, T ) be a compact space, with Seℓ(τV (T ),2)(X, T ) 6= ∅. Then, (a) (X, T ) is a linear ordered topological space (in particular, ind(X, T ) ≤ 1), (b) Tf = T for every f ∈ Seℓ(τV (T ),2)(X, T ), (c) SeℓτV (T )(X, T ) 6= ∅. Here, ind(X, T ) means the small inductive dimension of (X, T ). In view of Theorem 2.1, it makes some sense to investigate the topology Tf . For instance, the following simple observation was obtained in [16]. Proposition 2.3 ([16]). Let (X, T0) be a space, and let f ∈ Seℓ(τV (T0),2) (X, T0). Then, f ∈ Seℓ(τV (T ),2)(X, T ) for every topology T on X which is finer than T0. Consider the natural partial order on all Hausdorff topologies on a set X defined by T1 ≪ T2 provided T2 is finer than T1, i.e. T1 ⊂ T2. Then, by Proposition 2.3, f ∈ Seℓ(τV (T0),2) (X, T0) implies f ∈ Seℓ(τV (T ),2)(X, T ) for every Hausdorff topology T on X, with T0 ≪ T . Thus, we have the following natural question about a possible ≪-minimal topol- ogy T on a set X such that a given selection f ∈ Seℓ2(X) is τV (T )-continuous. Namely, Problem 2.4 ([16]). Let X be a set, and let f ∈ Seℓ2(X). Does there exist a topol- ogy T on X which is ≪-minimal with respect to the property “f ∈ Seℓ(τV (T ),2)(X, T )”? Some problems on selections for hyperspace topologies 73 Related to this question, let us observe that, by Theorem 2.1, f ∈ Seℓ(τV (T ),2)(X, T ) implies Tf ≪ T . So, Tf is a possible candidate for a ≪-minimal topology in that sense. However, we have the following recent example. Example 2.5 ([16]). There exists a set X and σ ∈ Seℓ2(X) such that σ is not τV (Tσ)-continuous. By Theorem 2.2, if (X, T ) is a compact space, then Seℓ(τV (T ),2)(X, T ) 6= ∅ if and only if SeℓτV (T )(X, T ) 6= ∅. On the other hand, Seℓ(τV (Te),2)(R, Te) 6= ∅, while SeℓτV (Te)(R, Te) = ∅ (see [8]), where R are the real numbers and Te is the usual Euclidean topology on R. Thus, in view of Theorem 2.1, we get the following natural question. Problem 2.6 ([16]). Let (X, T ) be a connected space, and let f ∈ Seℓ(τV (T ),2)(X, T ). Is it true that SeℓτV (T )(X, T ) 6= ∅ if and only if SeℓτV (Tf )(X, Tf ) 6= ∅? In general, the answer is “No” which was provided by the following example. Example 2.7 ([16]). There exists a separable, connected and metrizable space (X, T ) such that (i) Seℓ(τV (T ),2)(X, T ) 6= ∅, (ii) SeℓτV (Tf ) (X, Tf ) 6= ∅, for every f ∈ Seℓ(τV (T ),2)(X, T ), (iii) SeℓτV (T )(X, T ) = ∅. In contrast to this, Theorem 2.1 implies that Seℓ(τV (T ),n)(X, T ) = Seℓ(τV (T ),2)(X, T ), for every n ≥ 2, provided (X, T ) is a connected space. On this base, we have also the following question. Problem 2.8. Does there exist a space (X, T ) such that Seℓ(τV (T ),2)(X, T ) 6= ∅ but Seℓ(τV (T ),n)(X, T ) = ∅ for some n > 2? Suppose that (X, T ) is connected, and f ∈ Seℓ(τV (T ),2)(X, T ). Then, by Theorem 2.1, the space (X, Tf ) will be locally compact as a connected linear ordered space. Hence, a possible common point of view to Theorems 2.1 and 2.2 is suggested by the following question. Problem 2.9. Let (X, T ) be a locally compact space, with Seℓ(τV (T ),2)(X, T ) 6= ∅. Does there exist a topology T∗ ≪ T on X such that (X, T∗) is a linear ordered topological space? It should be mentioned that all known selection constructions are based on some extreme principle related to “orderability”. Hence, it seems natural to expect that some dimension-like function might be bounded. This is, in fact, the motivation for our next question. Problem 2.10. Does there exist a space (X, T ) such that Seℓ(τV (T ),2)(X, T ) 6= ∅ and ind(X, T ) > 1? For some related results and open questions we refer the interested reader to [2, 9, 11]. 3. On the cardinality of SeℓτV (T )(X, T ) The cardinality of SeℓτV (T )(X, T ) may provide some information for (X, T ) but mainly when it is finite. Theorem 3.1. For a space (X, T ), with SeℓτV (T )(X, T ) 6= ∅, the following holds: (a) If (X, T ) is connected, then |SeℓτV (T )(X, T )| ≤ 2, [18]. (b) SeℓτV (T )(X, T ) is finite if and only if (X, T ) has finitely many connected components, [22]. 74 Valentin Gutev and Tsugunori Nogura (c) If (X, T ) is infinite and connected, then |SeℓτV (T )(X, T )| = 2 if and only if (X, T ) is compact, [21]. For some other relations between |SeℓτV (T )(X, T )| and (X, T ), the interested reader is refer to [10, 21, 22]. 4. On the variety of SeℓτV (T )(X, T ) As it was mentioned above, all known selection constructions are based on some extreme principle, so our knowledge about particular members of SeℓτV (T )(X, T ) is mainly related to this. Here are some result about “extreme-like” members of SeℓτV (T )(X, T ). Theorem 4.1 ([17]). Let (X, T ) be a space, with SeℓτV (T )(X, T ) 6= ∅. Then, the set {f(X) : f ∈ SeℓτV (T )(X, T )} is dense in (X, T ) provided (X, T ) is zero-dimensional, while (X, T ) is totally disconnected provided {f(X) : f ∈ SeℓτV (T )(X, T )} is dense in (X, T ). Here, as usual, a space (X, T ) is zero-dimensional if it has a base of clopen sets, i.e. if ind(X, T ) = 0. Problem 4.2 ([17]). Does there exist a space (X, T ) which is not zero-dimensional but {f(X) : f ∈ SeℓτV (T )(X, T )} is dense in (X, T )? Problem 4.3. Let (X, T ) be a totally disconnected space, with SeℓτV (T )(X, T ) 6= ∅. Is the set {f(X) : f ∈ SeℓτV (T )(X, T )} dense in (X, T )? Some other results about extreme-like selections are summarized below. Theorem 4.4. For a space (X, T ), with SeℓτV (T )(X, T ) 6= ∅, the following holds: (a) (X, T ) is zero-dimensional provided for every point x ∈ X there exists an fx ∈ SeℓτV (T )(X, T ), with f −1 x (x) = {S ∈ F(X, T ) : x ∈ S}, [17]. (b) If (X, T ) is first countable and zero-dimensional, then for every point x ∈ X there exists an fx ∈ SeℓτV (T )(X, T ), with f −1 x (x) = {S ∈ F(X, T ) : x ∈ S}, [17]. (c) If (X, T ) is separable, then it is zero-dimensional and first countable if and only if for every point x ∈ X there exists an fx ∈ SeℓτV (T )(X, T ), with f−1x (x) = {S ∈ F(X, T ) : x ∈ S}, [10]. 5. More about the selection problem for topologically generated hyperspace topologies Suppose that “R” is a rule by which for any space (X, T ) we may assign a topology τR(T ) on F(X, T ) depending only on the topological structure T of X. We consider the class SeℓR of those spaces (X, T ) which admit a τR(T )-continuous selection for their hyperspaces F(X, T ) of closed subsets, i.e. (X, T ) ∈ SeℓR if and only if F(X, T ) has a τR(T )-continuous selection. Note that if “V ” is the rule by which we assign the Vietoris topology τV (T ) on F(X, T ), then (X, T ) ∈ SeℓV if and only if SeℓτV (T )(X, T ) 6= ∅. In what follows, let us recall that, for a space (X, T ), the Fell topology τF (T ) on F(X, T ) is defined by all basic Vietoris neighbourhoods 〈V〉 such that X \ ⋃ V is compact. As it becomes clear, we will use “F” to denote the rule that assigns the Fell topology. Under this terminology, some of the known results can be summarized as follows. Theorem 5.1. Let (X, T ) be a strongly zero-dimensional metrizable space. Then, (a) (X, T ) ∈ SeℓV if and only if (X, T ) is completely metrizable, [6, 8, 19]. Some problems on selections for hyperspace topologies 75 (b) (X, T ) ∈ SeℓF if and only if (X, T ) is locally compact and separable, [15]. The statement (b) of Theorem 5.1 is not surprising since the Fell topology τF (T ) on F(X, T ) is, in general, not admissible. Related to this, let us recall that a space (X, T ) is topologically well-orderable [8] if there exists a linear order ≺ on X such that (X, T ) is a linear ordered space with respect to “≺”, and every non-empty closed subset of (X, T ) has a “≺”-minimal element. For instance, a strongly zero- dimensional metrizable space (X, T ) is topologically well-orderable if and only if it is locally compact and separable, [8]. Theorem 5.2 ([14]). A space (X, T ) is topologically well-orderable if and only if (X, T ) ∈ SeℓF . A further generalization of Theorem 5.2 based on its proof was obtained in [1, 13]. 6. Selections in metrizable spaces Theorem 6.1 ([6, 8]). Let (X, T ) be a completely metrizable space such that dim(X, T ) = 0. Then, there exists a τV (T )-continuous selection for F(X, T ). Here, dim(X, T ) means the covering dimension of (X, T ). Most of the hypotheses in Theorem 6.1 are the best possible. A metrizable space (X, T ) is completely metrizable provided there exists a τV (T )-continuous selection for F(X, T ) [19] (see Theorem 5.1); The assumption dim(X, T ) = 0 cannot be dropped or even weakened to dim(X, T ) ≤ 1 [8, 20]. Related to this, the following question seems to be open. Problem 6.2. Does there exist a zero-dimensional metrizable space (X, T ) such that F(X, T ) has a τV (T )-continuous selection but dim(X, T ) > 0? 7. More continuous selections for metric-generated hyperspace topologies The continuity of a selection f ∈ SeℓτV (T )(X, T ) can be improved in several directions involving hyperspace topologies weaker than the Vietoris one. Towards this end, let us briefly recall some of the most important admissible hyperspace topologies on a metric space (X, d). In what follows, we use Td to denote the topology on X generated by a metric d on X. The Hausdorff topology τH(d) on F(X, Td) depends essentially on the metric d on X. It is the topology on F(X, Td) generated by the Hausdorff distance H(d) associated to d. Let us recall that H(d) is defined by H(d)(S, T ) = sup {d(S, x) + d(x, T ) : x ∈ S ∪ T } , S, T ∈ F(X, Td). It is well-known that τV (Td) coincides with τH(d) if and only if X is compact [18] while, in general, these two topologies are not comparable. In view of that, we need also some hyperspace topologies which are coarser than both τV (Td) and τH(d). A very interesting such topology is the d-proximal topology τδ(d) on F(X, Td) [4]. A base for τδ(d) is defined by all collections of the form 〈〈V〉〉d = { S ∈ 〈V〉 : d ( S, X\ ⋃ V ) > 0 } , where V is again a finite family of open subsets of (X, Td). Here, and in the sequel, we assume that d(S, ∅) = +∞ for every S ∈ F(X, Td). Another topology of this type is the d-ball proximal topology τδB(d) on F(X, Td). A base for τδB(d) is defined by all collections of the form 〈〈V〉〉d, where V is a finite family of open subsets of (X, Td) such that X\ ⋃ V is a finite union of closed balls of (X, d). A very similar to the d-ball proximal topology is the d-ball topology τB(d) on F(X, Td) generated by all collections of the form 〈V〉, where V runs over the finite 76 Valentin Gutev and Tsugunori Nogura families of open subsets of (X, Td) such that X\ ⋃ V is a finite union of closed balls of (X, d). Finally, we need also the Wijsman topology τW(d) which is the weakest topology on F(X, Td) such that all distance functionals d(x, ·) : F(X, Td) → R, x ∈ X, are continuous. It should be mentioned that τδ(d), τδB(d), τB(d) and τW(d) also depend on the metric d on X. However, they are metrizable only under additional conditions on the metric space (X, d). On the other hand, we always have the following (usually strong) inclusions τW(d) ⊂ τδB(d) ⊂ τδ(d) ⊂ τV (Td) ⋂ τH(d), and τδB(d) ⊂ τB(d) ⊂ τV (Td). For these and other properties of the above hyperspace topologies, we refer the interested reader to [3] and [4]. For a metrizable space (X, T ), let M(X, T ) denote the set of all metrics d on X compatible with the topology of X, i.e. for which Td = T . Concerning hyperspace topologies which are “mixed” – where the definition includes a topological part from the topological space (X, T ) and a metric part from a compatible metric on X, there arise at least three different points of view given by how useful selections for these hyperspace topologies are. Let τR be such a class of hyperspace topologies which are generated by the compatible metrics on X, i.e. for every d ∈ M(X, T ) we have a corresponding topology τR(d) on F(X, T ). For convenience, we will restrict our attention only to strongly zero-dimensional metrizable spaces considering the following: (S)w The class w-SeℓR of those strongly zero-dimensional metrizable spaces (X, T ) which have the Weak τR-Selection Property defined by (X, T ) ∈ w-SeℓR if and only if there exists a τR(d)-continuous selection for F(X, T ) for some d ∈ M(X, T ). (S) The class SeℓR of those strongly zero-dimensional metrizable spaces (X, T ) which have the τR-Selection Property defined by (X, T ) ∈ SeℓR if and only if F(X, T ) has a τR(d)-continuous selection for every d ∈ M(X, T ). (S)s The class s-SeℓR of those strongly zero-dimensional metrizable spaces (X, T ) which have the Strong τR-Selection Property defined by (X, T ) ∈ s-SeℓR if and only if F(X, T ) has a selection which is τR(d)-continuous for every d ∈ M(X, T ). Obviously, we always have s-SeℓR ⊂ SeℓR ⊂ w-SeℓR. However, in general, no one of these inclusions is invertible, see [15]. To become more specific, we will use R = W for the Wijsman topology; R = δB for the ball proximal topology; R = B for the ball topology; and R = δ for the proximal topology. Theorem 7.1 ([15]). In the class of strongly zero-dimensional metrizable spaces, the following holds: (a) SeℓF = s-SeℓW = SeℓW $ w-SeℓW . (b) SeℓF = s-SeℓδB = SeℓδB $ w-SeℓδB. (c) SeℓF = s-SeℓB $ SeℓB ⊂ w-SeℓB. (d) s-Seℓδ $ Seℓδ $ w-Seℓδ. Related the the above theorem, the following two questions are of interest. Problem 7.2 ([15]). Does there exist a strongly zero-dimensional non-separable metrizable space (X, T ) such that X ∈ w-SeℓR for some R ∈ {W, δB, B}? Some problems on selections for hyperspace topologies 77 Problem 7.3 ([15]). Does there exist a strongly zero-dimensional metrizable space (X, T ) such that X ∈ w-SeℓB\SeℓB? Finally, we have also the following two general questions: Problem 7.4 ([7, 15]). Let R ∈ {W, δB, B, δ}, (X, T ) be a strongly zero-dimen- sional metrizable space, and let d ∈ M(X, T ) be a compatible metric. Does there exist a topological property P such that F(X, T ) has a τR(d)-continuous selection if and only if (F(X, T ), τR(d)) ∈ P? Problem 7.5 ([7, 15]). Let R ∈ {W, δB, B, δ}, (X, T ) be a strongly zero-dimen- sional metrizable space, and let d ∈ M(X, T ) be a compatible metric. Does there exist a metric property D such that F(X, T ) has a τR(d)-continuous selection if and only if d ∈ D? The interested reader is referred to [5, 7, 12, 15] for some additional discussion on the topic. 8. Special metrics and selections Let (X, d) be a metric space. A subset A ⊂ X is called d-clopen if d(A, X\A) > 0, [7, 15]. Every d-clopen set is clopen but the converse fails. For more information about this concept, see [7, 15]. We shall say that a metric space (X, d) is totally disconnected with respect to d, or totally d-disconnected, if every singleton of X is an intersection of d-clopen subsets of (X, d), [7]. Example 8.1 ([7]). There exists a metric space (X, d) with only two non-isolated points which is not totally d-disconnected. In view of this example, the following questions about the selection problem for the d-proximal topology are still open. Problem 8.2 ([7]). Let (X, T ) be a (strongly zero-dimensional) completely metriz- able space, and let d ∈ M(X, T ) be such that (X, d) is totally d-disconnected. Does there exist a τδ(d)-continuous selection for F(X, T )? Problem 8.3 ([7]). Let X be a metrizable scattered space, and let d ∈ M(X, T ) be such that (X, d) is totally d-disconnected. Does there exist a τδ(d)-continuous selection for F(X, T )? Problem 8.4 ([7]). Let (X, T ) be a metrizable scattered space, and d ∈ M(X, T ). Does there exist a τδ(d)-continuous selection for F(X, T )? The above question is open even in the special case when (X, T ) has only two non-isolated points, The answer is “Yes” if (X, T ) has only one non-isolated point [7, Theorem 5.5]. Finally, the following further question seems to be also interesting. Problem 8.5. Let (X, T ) be a metrizable space which is scattered with respect to compact subsets, i.e. every non-empty closed subset of (X, T ) contains a non-empty compact and relatively open subset. Also, let d ∈ M(X, T ). Does there exist a τδ(d)- continuous selection for F(X, T )? References [1] G. Artico and U. Marconi, Selections and topologically well-ordered spaces, Topology Appl. 115 (2001), 299–303. [2] G. Artico, U. Marconi, J. Pelant, L. Rotter, and M. Tkachenko, Selections and suborderability, Fund. Math. 175 (2002), no. 1, 1–33. 78 Valentin Gutev and Tsugunori Nogura [3] G. Beer, Topologies on closed and closed convex sets, Mathematics and its applications, vol. 268, Kluwer Academic Publishers, The Netherlands, 1993. [4] G. Beer, A. Lechicki, S. Levi, and S. Naimpally, Distance functionals and suprema of hyperspace topologies, Ann. Mat. Pure Appl. 162 (1992), 367–381. [5] D. Bertacchi and C. Costantini, Existence of selections and disconnectedness properties for the hyperspace of an ultrametric space, Topology Appl. 88 (1998), 179–197. [6] M. Choban, Many-valued mappings and Borel sets. I, Trans. Moscow Math. Soc. 22 (1970), 258– 280. [7] C.Costantini and V. Gutev, Recognizing special metrics by topological properties of the “metric”- proximal hyperspace, Tsukuba J. Math. 26 (2002), no. 1, 145–169. [8] R. Engelking, R. W. Heath, and E. Michael, Topological well-ordering and continuous selections, Invent. Math. 6 (1968), 150–158. [9] S. Fujii and T. Nogura, Characterizations of compact ordinal spaces via continuous selections, Topology Appl. 91 (1999), 65–69. [10] S. Garćıa-Ferreira, V. Gutev, T. Nogura, M. Sanchis, and A. Tomita, Extreme selections for hy- perspaces of topological spaces, Topology Appl. 122 (2002), 157–181. [11] S. Garćıa-Ferreira and M. Sanchis, Weak selections and pseudocompactness, preprint, 2001. [12] V. Gutev, Selections and hyperspace topologies via special metrics, Topology Appl. 70 (1996), 147–153. [13] , Fell continuous selections and topologically well-orderable spaces II, Proceedings of the Ninth Prague Topological Symposium (2001), Topology Atlas, Toronto, 2002, pp. 157–163 (elec- tronic). [14] V. Gutev and T. Nogura, Fell continuous selections and topologically well-orderable spaces, In- ternal Report No. 13/99, University of Natal. [15] , Selections for Vietoris-like hyperspace topologies, Proc. London Math. Soc. 80 (2000), no. 3, 235–256. [16] , Selections and order-like relations, Applied General Topology 2 (2001), 205–218. [17] , Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc. 129 (2001), 2809–2815. [18] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. [19] J. van Mill, J. Pelant, and R. Pol, Selections that characterize topological completeness, Fund. Math. 149 (1996), 127–141. [20] J. van Mill and E. Wattel, Selections and orderability, Proc. Amer. Math. Soc. 83 (1981), no. 3, 601–605. [21] T. Nogura and D. Shakhmatov, Characterizations of intervals via continuous selections, Rendi- conti del Circolo Matematico di Palermo, Serie II, 46 (1997), 317–328. [22] , Spaces which have finitely many continuous selections, Bollettino dell’Unione Matematica Italiana 11-A (1997), no. 7, 723–729. Received October 2002 Accepted December 2002 Valentin GUTEV (gutev@nu.ac.za) School of Mathematical and Statistical Sciences, Faculty of Science, University of Natal, King George V Avenue, Durban 4041, South Africa Tsugunori NOGURA (nogura@ehimegw.dpc.ehime-u.ac.jp) Department of Mathematics, Faculty of Science, Ehime University, Matsuyama, 790-8577, Japan