EstyAGT.dvi @ Applied General Topology c© Universidad Politécnica de Valencia Volume 8, No. 2, 2007 pp. 259-265 CL(R) is simply connected under the Vietoris topology N. C. Esty Abstract. In this paper we present a proof by construction that the hyperspace CL(R) of closed, nonemtpy subsets of R is simply connected under the Vietoris topology. This is useful in considering the conver- gence of time scales. We also present a construction of the (known) fact that this hyperspace is also path connected, as part of the proof. 2000 AMS Classification: 54B20, 54D05 Keywords: hyperspace, Vietoris topology, simply connected, path con- nected, time scales 1. Introduction Spaces of all non-empty and closed subsets of a topological space (or hy- perspaces) are a critical part of the study of time scales. The theory of time scales attempts to organize the solution methods for differential and difference equations which, when considered under the same equation, sometimes have very similar solutions and sometimes have wildly different solutions. The ap- proach is to consider a dynamic equation over an unknown domain, which is a non-empty and closed subset of R, or in other words, a point in CL(R). Such points of CL(R) are called time scales. For a good introduction to the theory of time scales, see [2]. When studying time scales in this context, there are immediate and interesting questions involving convergence: If a sequence of time scales converges, and we consider solutions of the same dynamic equa- tion over each member of the sequence, will the solutions converge? Of course, a formalized concept for convergence of functions over different domains is needed (in addition to a formalized concept of “sameness”). Results for some of these questions have been given in [12], when the solutions are unique and the dynamic equation is sufficiently continuous. 260 N. C. Esty Central to this discussion is the topology on the space of time scales. There are several well known topologies on hyperspaces, including the Hausdorff met- ric topology and the Vietoris topology. The Hausdorff metric topology of CL(R) is, happily, metrizable, but it has the unfortunate property that un- der it, [−n, n] does not converge to R. Since this convergence would be useful in the context of time scales, we turn instead to the Vietoris topology. This topology is not metrizable on CL(R); however, on hyperspaces associated to compact metrizable spaces it coincides with the Hausdorff metric topology. In 1951, it was shown by Michael that CL(R) was completely regular, seper- able, and first countable; see [14]. The statement that it is locally compact turned out to contain an error – a correction to the problematic proposition can be found in [5] – however it is now known that CL(R) is not locally compact. A result of Ivanova, Keesling and Velichko says that if the Vietoris topology on CL(X) is normal, then X is compact: see [10], [11], and [16]. It follows that CL(R) is not a normal space. In 2003, Hola, Pelant and Zsilinszky showed that CL(R) is not developable and that it is submetrizable; see [8]. It is also known to be strong alpha-favorable; this follows from statements in [19]. More attention has been paid to hyperspaces in the case that X is compact. It was shown as far back as 1931 by Borsuk and Mazurkiewicz that for a metrizable continuum X, both the hyperspace K(X) of compact subsets of X and C(X), the hyperspace of subcontinua of X, are path connected [4]. The non-metrizable case was investigated by McWaters [13] and Ward [17]. Local path connectedness of K(X) and C(X) was shown to be equivalent to local connectedness of X if X is compact in [18] in 1939. For topological properties on compact Vietoris hyperspaces, see the 1978 book of Nadler [15], or the more recent book by Illanes and Nadler, [9]. In 2002, in [6], Costantini and Kubis showed that under the Vietoris topol- ogy, CL(R) is pathwise connected but not locally connected. They actually showed a stronger statement, applying to a wider class of topologies, and giv- ing conditions for path-wise connectedness. They also give several results for the hyperspace of closed, bounded sets under the Hausdorff metric topolgoy, including that it is an absolute retract. For the reader who is not familiar with it, in Section 2 we will briefly discuss the Vietoris topology on a general topological space X. Then in Section 4 we shall prove: Theorem 1.1. Under the Vietoris topology, CL(R) is simply connected. For the purposes of the proof, we will also present an alternate proof that CL(R) is path connected, by constructing an explicit path from any point in CL(R) to R; this will be done in Section 3. This will assist in the construction of a nullhomotopy of an arbitrary loop in Section 4. CL(R) is simply connected under the Vietoris topology 261 2. The Vietoris topology Suppose that you have a topological space (X, τ ). The Vietoris topology is one of a group of topologies called “hit and miss” topologies. The name is indicative of the fact that open sets in the space CL(X) are given by those subsets of X which “hit” certain specific open sets of X and “miss” their complements. For a full discussion of hit and miss topologies, see [1]. Let U1, . . . , Un be a finite collection of open sets in X, i.e. members of τ . We define an open set in CL(X), denoted B =< U1, . . . , Un >, to be all those non-empty and closed subsets A of X satisfying the following two properties: (1) A ∩ Ui 6= ∅, for i = 1, . . . , n. (“hit”) (2) A ⊂ ⋃n i=1 Ui (“miss”) The collection of all such sets, for any finite collection of Ui, forms a basis for the Vietoris topology on CL(X). When X = R, it is not hard to see that under this topology, the sequence of time scales Tn = [−n, n] does in fact converge to R. An alternative way of looking at the Vietoris topology is to use the fact that it is the supremum of the upper and lower Vietoris topologies, the first of which is generated by all sets of the form U + = {A ∈ CL(X) : A ⊂ U}, and the second of which is generated by sets of the form U − = {A ∈ CL(X) : A ∩ U 6= ∅}, where U is a τ -open set. Subbase elements of the Vietoris topology are of the form U + with U ∈ τ and ⋂ U∈U U −, with U ⊂ τ finite. 3. Path connected In the following, we will consider CL(R) endowed with the Vietoris topology. Theorem 3.1. CL(R) is path connected. Proof. Let T ∈ CL(R) be an arbitrary point of the hyperspace. In the future, to distinguish between points of CL(R) and points of R, we will refer to the former as time scales. We construct a path from T to R. As T 6= ∅, choose t0 ∈ T. Define γ : [0, 1] → CL(R) by γ(1) = R, and for s ∈ [0, 1), γ(s) = T ∪ [t0 − s 1 − s , t0 + s 1 − s ] We denote by A(s) the closed interval [t0 − s 1−s , t0 + s 1−s ]. Note that γ(s) is clearly nonempty and closed, as it is the finite union of closed sets, the first of which is always nonempty. Note also that lims→1 s 1−s diverges to infinity. 262 N. C. Esty First we must show that γ is continuous. It is enough to show γ is continuous with respect to the upper and lower Vietoris topologies. First let γ(s0) ∈ U +, where U + is a basic open set in the upper Vietoris topology. Then γ(s0) ⊂ U . If s0 = 1, then γ(s0) = R ⊂ U = R, so clearly for all s ∈ [0, 1], γ(s) ⊂ U and γ(s) ∈ U +. Assume s0 6= 1. As U is open and γ(s0) is compact, there exists some ǫ > 0 such that B(γ(s0), ǫ) ⊂ U . By continuity of f (x) = x 1−x , there exists some δ > 0 such that if |s − s0| < δ, then |f (s) − f (s0)| < ǫ, and therefore A(s) ⊂ U , so γ(s) ∈ U +. Next suppose γ(s0) ∈ U − 1 ∩ · · · ∩ U − n , a basic open set in the lower Vietoris topology. If s0 = 1, then there is some ǫ > 0 such that if s ∈ (1 − ǫ, 1], A(s)∩Ui 6= ∅ for all i. In addition, if T ∈ U − i , then γ(s) ∈ U − i for all s ∈ [0, 1]. So assume that T /∈ U − i and s0 6= 1. Therefore A(s0) ∈ U − 1 ∩ · · · ∩ U − n . Choose ti ∈ A(s0) ∩ Ui, and let di > 0 be such that B(ti, di) ⊂ Ui ∩ A(s0) for all i ∈ {1, . . . , n}. As f is continuous, we can find δ > 0 such that if |s − s0| < δ, then |f (s) − f (s0)| < min{d1, . . . , dn}. Then ti ∈ γ(s) for all i and therefore γ(s) ∈ U −1 ∩ · · · ∩ U − n . � 4. Simply connected It is enough to show that all loops with a particular base point T are null- homotopic. We choose the base point to be R, and assume that we are given an arbitrary loop based at R, i.e. a continuous map f : [0, 1] → CL(R) with f (0) = f (1) = R. Lemma 4.1. There exists a continuous map x : [0, 1] → R such that x(s) ∈ f (s). Proof. We define x : [0, 1] → R by letting x(s) be the point of f (s) which is closest to the origin, choosing the positive point in the case of a tie. This map is well-defined because each f (s) ∈ CL(R), meaning it is a nonempty and closed subset of the real line, so such a point exists. We claim that this map is in fact a loop in R. It is easy to see that x(0) = x(1) = 0, so we need only check continuity. Notationally we will sometimes write xs for x(s). Fix s0 ∈ [0, 1] and fix ǫ > 0. Consider f (s0). We know that x(s0) ∈ f (s0) by definition of x. Consider the ball around f (s0) in CL(R) B1 =< R, (xs0 − ǫ/2, xs0 + ǫ/2) > Continuity of f implies there exists a δ1 > 0 such that if s ∈ (s0 −δ1, s0 +δ1), then f (s) ∈ B1. In particular, f (s) ∩ (xs0 − ǫ/2, xs0 + ǫ/2) 6= ∅. Therefore the closest point of f (s) to the origin can have distance from the origin no greater than |xs0 | + ǫ/2. CL(R) is simply connected under the Vietoris topology 263 Should xs0 be within ǫ/2 of the origin, we can let δ = δ1 at this point. If not, consider B2 =< (−∞, −|xs0| + ǫ/2), (|xs0| − ǫ/2, ∞) > Because f (s0) contains no points closer to the origin than xs0 , f (s0) ∈ B2. By continuity of f , there exists some δ2 > 0 such that s ∈ (s0 − δ2, s0 + δ2) implies f (s) ∈ B2. But this means that the closest point of f (s) to the origin can have distance from the origin no less than |xs0 | − ǫ/2. Choose δ = min{δ1, δ2}. Then for all s ∈ (s0 − δ, s0 + δ), the closest point of f (s) to the origin lies within ǫ/2 of either xs0 or −xs0 . By choosing the positive one in all tie cases, we ensure that in fact xs is within ǫ/2 of xs0 . Therefore x is a continuous function. � Given an arbitrary point p0 ∈ T, let γT : [0, 1] → CL(R) be the map defined by γT(s) = T ∪ [p0 − s 1−s , p0 + s 1−s ] for s ∈ [0, 1) and γT(1) = R. We know from the proof of Theorem 3.1 in Section 3 that this map is a continuous path from T to R. Since γ depends on the point p0, we will sometimes write γT(p0, s). We wish to find a homotopy from f to the constant loop c(s) = R, s ∈ [0, 1]. In other words, we require a continuous map F : [0, 1] × [0, 1] → CL(R) with the following properties: (1) For all s ∈ [0, 1], F (s, 0) = f (s), i.e., at time zero we have the original loop f . (2) For all s ∈ [0, 1], F (s, 1) = R, i.e., at time one we have the constant loop c. (3) For all t ∈ [0, 1], F (0, t) = F (1, t) = R, i.e., at all other times we do, in fact, have loops based at R. Theorem 4.2. F (s, t) = γf (s)(x(s), t) is a homotopy from f to the constant loop. Proof. It is easy to see that F has the three properties listed. Continuity is all that remains to check. Fix a particular point (s0, t0). It is clear that F is continuous in t because the path γf (s0) is continuous. Let us check continuity in s. Note that if t = 1, then F (s, t) = R for all s. Therefore we need only consider the case t0 6= 1. Again, we check continuity with respect to the upper and lower Vietoris topologies. We know that F (s0, t0) = fs0 ∪ [xs0 − t0 1−t0 , xs0 + t0 1−t0 ] For brevity, we will refer to that closed interval as Is0 . 264 N. C. Esty Let F (s0, t0) ∈ U +, a basic open set in the upper Vietoris topology. As f is continuous, there exists some δ1 > 0 such that if |s − s0| < δ1, then fs ∈ U +. Because Is0 is compact, there is some ǫ > 0 such that B(Is0 , ǫ) ⊂ U . By continuity of x(s), there exists some δ2 such that if |s − s0| < δ2, then |xs − xs0| < ǫ. Then it is clear that Is ⊂ B(Is0 , ǫ), and so Is ∈ U +. Let |s − s0| < min{δ1, δ2}, and we have that F (s, t0) ∈ U +. Next let F (s0, t0) ∈ U − 1 ∩ · · · ∩ U − n , a basic open set in the lower Vietoris topology. If fs0 ∈ U − i , then by continuity of f , there exists some δ > 0 such that if |s − s0| < δ, fs ∈ U − i , and so F (s, t0) ∈ U − i . So we can suppose without loss of generality that fs0 /∈ U − i for all i. Therefore Is0 ∈ U − 1 ∩ · · · ∩ U − n . We use a reasoning similar to that in the proof of Theorem 3.1. Take ti ∈ Is0 ∩ Ui, and let di such that B(ti, di) ⊂ Is0 ∩ Ui. There exists some δ > 0 such that when |s − s0| < δ, |xs − xs0| < min{d1, . . . , dn} and therefore ti ∈ Is for all i. Thus we have that F (s, t0) ∈ U − 1 ∩ · · · ∩ U − n . � Acknowledgements. I would like to thank Drs. Bonita Lawrence and Ralph Oberste-Vorth for suggesting this problem. References [1] G. Beer and R. K. Tamaki, On hit-and-miss hyperspace topologies, Comment. Math. Univ. Carolinae 34 (1993), 717–728. [2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhauser; 2001. [3] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser; 2003. [4] K. Borsuk and S. Marzurkiewicz, Sur l’hyperspace d’un continu, C. R. Soc. Sc. Varsovie 24 (1931), 142–152. [5] C. Constantini, S. Levi and J. Pelant, Compactness and local compactness in hyper- spaces, Topology Applications 123 (2002), 573–608. [6] C. Constantini and W. Kubis, Paths in hypserspaces , App. Gen. Top. 4 (2003), 377–390. [7] D. W. Curtis, Hyperspaces of noncompact metric spaces, Comp. Math. 40 (1980), 139– 152. [8] L. Hola, J. Pelant and L. Zslinszky, Developable hyperspaces are metrizable, App. Gen. Top. 4 (2003), 351–360. [9] A. Illanes and S. Nadler, Hyperspaces, Marcel-Dekker (1999). [10] V. M. Ivanova, On the theory of the space of subsets, Dokl. Akad. Nauk. SSSR 101 (1955), 601–603. [11] J. Keesling, On the equivalence of normality and compactness in hyperspaces, Pacific J. Math. 33 (1970), 657–667. [12] B. Lawrence and R. Oberste-Vorth, Solutions of dynamic equations with varying time scales, Proc. Int. Con. of Difference Equations, Special Functions and Applications (2006). CL(R) is simply connected under the Vietoris topology 265 [13] M. M. McWaters, Arcs, semigroups and hyperspaces, Can. J. Math. 20 (1968), 1207– 1210. [14] E. Michael, Topologies on spaces of subsets, Trans. Am. Math. Soc. 71 (1951), 152–182. [15] S. Nadler, Hyperspaces of Sets, Marcel-Dekker (1978). [16] N. V. Velichko, On spaces of closed subsets, Sibirskii Matem. Z. 16 (1975), 627–629. [17] L. E. Ward, Arcs in hyperspaces which are not compact, Proc. Amer. Math. Soc. 28 (1971), 254–258. [18] M. Wojdyslawski, Retractes absolus et hyperespaces des continus, Fund. Math. 32 (1939), 184–192. [19] L. Zsilinszky, Topological games and hyperspace topologies, Set-Valued Anal. 6 (1998), 187–207. Received April 2006 Accepted September 2006 N. C. Esty (esty@marshall.edu) Department of Mathematics, Marshall University, One John Marshall Drive, Huntington, WV, 45669, USA