Houston Journal of Mathematics c 2000 University of Houston ­ Volume 26, No. 4, 2000 ON A SPECIAL METRIC YUN ZIQIU AND HEIKKI J.K. JUNNILA Communicated by Jun-iti Nagata Abstract. In this note, we prove that whenever d is a compatible metric for a hedgehog space J having more than 2c spines, there exists ¯ > 0 and x J such that the family B (y; ¯): y B (x; ¯) contains more than 2 f d 2 d g c distinct sets. This result provides a negative answer to a question raised by Nagata in [6]. We also give positive answers to the same question under some extra conditions. 1. Introduction J. Nagata proved in [5] that each metrizable space X has a compatible metric d such that for every ¯ > 0, the family B (x; ¯): x X of all ¯-balls of (X; d) f d 2 g is closure-preserving, and in [6] he raised the following question: Does there exist, on each metrizable space X, a compatible metric d such that for every ¯ > 0, the family B (x; ¯): x X of all ¯-balls of (X; d) is hereditarily f d 2 g closure-preserving (as an unindexed family)? In this note, we answer the above question negatively by proving that there exists no compatible metric d on the hedgehog space H = J((2c)+) such that for every ¯ > 0, the family B (x; ¯): x H of all ¯-balls is point ¬nite or even f d 2 g point countable (as an unindexed family). We also show that the answer to the above question is positive under some added restrictions. Notation and terminology. A metric d of a (metrizable) topological space X is a compatible metric if the family B (x; ¯): x X; ¯ > 0 of all open balls is a base f d 2 g for the topology of X, where Bd(x; ¯) = y X : d(x; y) < ¯ . Denote by I the f 2 g 1991 Mathematics Subject Classi¬cation. 54E35; 54E99. Key words and phrases. Compatible metric, hedgehog. While this paper was written, the ¬rst author was visiting the University of Helsinki and he was supported by the Magnus Ehrnrooth Foundation in Finland and the Education Committee of Jiangsu Province in China. 877 878 YUN ZIQIU AND HEIKKI JUNNILA closed unit interval with the ordinary euclidean topology, and for every ordinal number «, let I« be the copy of I obtained by setting I« = I « . We denote by ¢ f g H the Hilbert cube ¥«<!I« { a Cartesian product of countably in¬nitely many copies of I. Let ´ be an in¬nite cardinal number. The hedgehog J(´) with ´ spines ([9] and [3]) is the (metrizable) space whose underlying set is obtained from the union I« : « < ´ by identifying all the zero points (0;«) into one point, which f g we denote by 0, and whose topology is de¬ned as follows: for every « < ´, the S set I« (0;«) is open in J(´) and has the same relative topology in J(´) as n f g in I«; the point 0 has the sets 0 (r; «) J(´) : 0 < r < ¯ , for ¯ > 0, as f g [ f 2 g a neighborhood base. A family of subsets of a topological space X is closure- U preserving if for each subfamily 0 of , Cl U : U 0 = ClU : U 0 . U U f 2 U g f 2 U g The family is hereditarily closure-preserving, if for any choice of sets L U, U S S U ² for U , the family L : U is closure-preserving. It is well known and 2 U f U 2 Ug easy to prove that every locally ¬nite family is hereditarily closure-preserving. The symbol c denotes the cardinality of the continuum. For a set S, the symbol [S]2 is used to denote the family consisting of all two-element subsets of a set S. For terms not de¬ned here, refer to [2]. 2. The Results The following lemma contains just the special case \« = c" of the ErdÁos-Rado Theorem (2«)+ («+)2 (see e.g., [8], p.9). ! « Lemma 2.1. (ErdÁos-Rado) Let S be a set of cardinality S = (2c)+. If is a j j P partition of [S]2 with c, then there exist A S and P such that A > c jPj ´ ² 2 P j j and [A]2 P . ² Proposition 2.2. Let ´ = (2c)+, and let d be a compatible metric for the hedge- hog space J(´). Then there exists ¯ > 0 and x J(´) such that the family 2 B (y; ¯): y B (x; ¯) contains more than c distinct sets. f d 2 d g Proof. For every (r; «) J(´) 0 , denote the point (r; «) by r . For each 2 n f g « « < ´, let 0« = 0. Denote by Q the set of all rational numbers in I. To prove the proposition, assume on the contrary that there exists a compatible metric d on J(´) such that for all ¯ > 0 and x J(´), the family B (y; ¯): y 2 f d 2 B (x; ¯) contains at most c distinct sets. For each « < ´, de¬ne a function d g f« : Q Q [0; ) by setting f«(y; z) = d(y«; z«) for y«; z« I« with y; z Q. ¢ ! 1 2 2 Since there are only c! = c di«erent functions Q Q [0; ), we can ¬nd a ¢ ! 1 set K ´ with cardinality ´ such that for all «; ¬ K, f = f . Without loss ² 2 « ¬ of generality, we may assume that K = ´. We then have, for all y; z Q, that 2 SPECIAL METRIC 879 d(y ; z ) = d(y ; z ) for any pair «; ¬ ´. Since d is continuous, the following « « ¬ ¬ 2 stronger condition holds: d(y«; z«) = d(y¬; z¬) for all «; ¬ ´ and y; z I : (1) 2 2 For all «; ¬ ´ with « = ¬, de¬ne g : Q Q [0; ) by setting g (y; z) = 2 6 «;¬ ¢ ! 1 «;¬ d(y ; z ) for all y; z Q. Since there are only c! = c functions Q Q [0; ), « ¬ 2 ¢ ! 1 we can use Lemma 1 to ¬nd a set L ´ with cardinality bigger than c such that ² for all «; ¬; «0; ¬0 L with « = ¬ and «0 = ¬0 we have that g = g . We 2 6 6 «;¬ «0;¬0 then have, for all y; z Q, that d(y ; z ) = d(y ; z ) for all «; ¬; «0; ¬0 L with 2 « ¬ «0 ¬0 2 « = ¬ and «0 = ¬0. By continuity of d, the following holds: 6 6 d(y«; z¬) = d(y« ; z¬ ) if y; z I; «; ¬; «0; ¬0 L; « = ¬ and «0 = ¬0 : (2) 0 0 2 2 6 6 Let y I. Note that it follows from Condition (2) that, whenever « = ¬, the 2 6 distance d(y«; I¬) = min d(y«; z¬): z I is independent of «; ¬ L. Hence f 2 g 2 we can denote by D(y) the number d(y«; I¬), for «; ¬ L, « = ¬. Note that 2 6 D(y) > 0 whenever y > 0. Claim. For each y (0; 1], there is ¯ > 0 such that z (y ¯; y + ¯) implies 2 2 − D(z) D(y). To prove the Claim, note that B (y ;D(y)) is a neighborhood of ´ d ! y and that there thus exists ¯ > 0 such that (y ¯; y + ¯) ! B (y ;D(y)). ! − ¢ f g ² d ! By Condition (1), we have that (y ¯; y + ¯) « B (y ;D(y)) for every − ¢ f g ² d « « L. We show that if z (y ¯; y + ¯), then D(z) D(y). Assume that this 2 2 − ´ is not true, and let z (y ¯; y + ¯) be such that D(z) > D(y). Let « L. 2 − 2 Note that there exists w [0; 1] such that d(y ; w ) = D(y) for each ¬ L « . 2 « ¬ 2 n f g Since D(z) > D(y), we have that w B (y ;D(z)); by condition (2) it follows ¬ 2 d « that w B (y ;D(z)) for each ¬ L « . Note that, for all ¬; ¬0 L, ¬0 = ¬, « 2 d ¬ 2 n f g 2 6 we have that d(z ; y ) D(z), and hence that z = B (y ;D(z)); on the other ¬ ¬0 ≥ ¬ 2 d ¬0 hand, we have that z (y ¯; y + ¯) ¬ B (y ;D(y)) B (y ;D(z)). The ¬ 2 − ¢ f g ² d ¬ ² d ¬ foregoing shows that B (y ;D(z)) = B (y ;D(z)) whenever ¬; ¬0 L, ¬ = ¬0. d ¬ 6 d ¬0 2 6 As a consequence, the subfamily B (y ;D(z)) : ¬ L of B (y ;D(z)) : y f d ¬ 2 g f d ¬ ¬ 2 B (w ;D(z)) contains L > c distinct sets in contradiction to our assumption. d « g j j Hence the Claim is proved. Note that D is a continuous function I R. The Claim shows that, for each ! y (0; 1], there is an ¯ > 0, such that D(y) is the maximum value in (y ¯; y + ¯). 2 − By an elementary argument of mathematical analysis, we can prove that D(y) is constant on (0; 1]. It follows that D(y) = D(0) = 0 for every y I, and this is 2 clearly impossible. 880 YUN ZIQIU AND HEIKKI JUNNILA Every hedgehog space is a ¬rst countable T1-space without isolated points, and hence Corollary 3 of [1] shows that every hereditarily closure-preserving open family in a hedgehog space is locally ¬nite; as a consequence, we have the following solution to Nagata's problem. Corollary 2.3. For ´ = (2c)+, the hedgehog J(´) has no compatible metric d such that for every ¯ > 0, the family of ¯-balls B (x; ¯): x J(´) is hereditarily f d 2 g closure-preserving (as an unindexed family). In the following, we show that under certain added conditions the answer to Nagata's question is positive.