A Hahn-Mazurkiewicz Theorem for generalized Peano continua

A Hahn-Mazurkiewicz Theorem for generalized Peano continua By R. AYALA,M.J.CHA ÂVEZ and A. QUINTERO Abstract. S. Mazurkiewicz remarked after his proof of the celebrated HahnMazurkiewicz Theorem ([5]) that any generalized Peano continuum is the continuous image of the half-line 0;1 but the converse does not hold. Therefore continuous images of 0;1do not characterize generalized Peano continua. In this paper we use perfect maps and trees to obtain an analogue of the Hahn-Mazurkiewicz Theorem for generalized Peano continua in the spirit of the classical Hahn-Mazurkiewicz Theorem. 0. Introduction. The well known Hahn-Mazurkiewicz Theorem ([5]) establishes that a Hausdorff space Pis a Peano continuum if and only if it is the image of the unit interval I0;1. This theorem was the culmination of the study started by Peanos celebrated example of a square-filling curve. When compactness is replaced by local compactness we still have an interesting class of topological spaces, namely, generalized Peano continua. These spaces were already considered by the founders of continuum theory. In fact, Mazurkiewicz in his seminal paper [5] showed that any generalized Peano continuum is the continuous image of the halfline 0;1 but the converse does not hold. Therefore continuous images of 0;1 do not characterize generalized Peano continua. In this paper we use perfect maps and trees to obtain an analogue of the Hahn-Mazurkiewicz Theorem for generalized Peano continua (Theorem 2.4). In spite of its simple nature this result seems to be new in the literature. We use this theorem to give two further results on generalized continua which extend two basic results on Peano continua (Corollaries 2.6 and 2.7). 1. Generalized Peano continua. In this paper we shall deal with the class of generalized Peano continua. We recall that a continuum Xis a compact connected metrizable space. When compactness is replaced by local compactness the space Xis called a generalized continuum. If in addition Xis locally connected it is called a (generalized) Peano continuum. Hence, any (generalized) Peano continuum is arcwise connected by ([7]; 4.2.5). Moreover it follows from ([3]; 4.4 F.(c)) that any generalized continuum is separable and hence second countable and s-compact ( [3]; 4.1.16, and 3.8.c(b) ). The local compactness together with the s-compactness yield that Xis a countable union [ 1 n1Knof compact subsets Kn7Xwith Kn7int Kn1. Actually we can assume without loss of generality that each Knis connected and all the components of XÿKnare unbounded. Indeed, each Knis contained in a finite Mathematics Subject Classification (1991): 54F15, 54E40. union of open connected subsets of compact closure K0 n.IfK0 nis not connected we can consider a new K0 nby adding to K0 ncompact and connected subspaces joining its (finite) components. If some components of XÿKnare bounded then we consider a new K00 nby adding to Knall the bounded components in XÿKn. The sequence fKngn^1with these two properties will be called an exhausting sequence in X. Given an exhausting sequence in XfKngn^1aFreudenthal end of Xis a sequence " Cnn^1of components Cn7XÿKnwith Cn17Cn. We denote by fXthe set of Freudenthal ends of X. The set b XX[fXadmits a compact topology whose basis consists of the open sets of Xtogether with the sets c CnCn[ f"2fX;Cnappears in "g n^1. This topology (which does not depend on the sequence fKngn^1) is called the Freudenthal topology and b Xis called the Freudenthal compactification of X. Moreover the subspace fXturns out to be homeomorphic to a closed subset of the Cantor set (see ([4]) for details). Aperfect map f :Xÿ!Yis a continuous closed map such that for each y2Ythe fiber fÿ1yis compact. If Xand Yare locally compact Hausdorff spaces fis perfect if and only if it is continuous and fÿ1Kis compact for any compact subset K7Y(i.e. fis a proper map ([1]; I.10.2.1)). If Xand Yare generalized continua any perfect (or equivalently proper) map f:Xÿ!Y extends to a continuous map b f: b Xÿ! b Ywhich restricts to a continuous map f:fX ÿ! fY. Namely if " Cnn^1; b f"  f"  Dkk^1where fCnk  Dk for some increasing subsequence Cnkk^1of ". The classical Hahn-Mazurkiewicz Theorem establishes that a Hausdorff space Pis a Peano continuum if and only if it is the continuous image of the unit interval 0;1. In particular Peano continua are preserved by continuous maps. In addition to the above theorem S. Mazurkiewicz ([5]) also shows that any generalized Peano continuum is the continuous image of the half-line 0;1  but the converse does not hold; that is, generalized Peano continua are not preserved by continuous maps. In fact they are preserved by perfect maps. Namely Lemma 1.1. Let X be a generalized Peano continuum and f :Xÿ!Y a perfect surjection. Then Y is a generalized Peano continuum. Proof. Indeed, perfect maps preserve all properties which define Peano continua (see [3]; 4.4.15, [3]; 3.7.21, and [7]; 3.5.7(4)). h Furthermore the following lemma shows that most of generalized Peano continua are not perfect images of 0;1 . In particular only one-ended spaces can be perfect images of 0;1 . Lemma 1.2. Let X and Y be a generalized Peano continua. Any perfect surjection f:Xÿ!Y induces a continuous surjection b f: b Xÿ! b Y and f:fXÿ!fY. Proof. Since b Xand b Yare Hausdorff compact spaces b fis a closed map and hence fX  b Y7 b f b X. That is, b fand fare onto maps. h 2. An analogue of the Hahn-Mazurkiewicz Theorem for generalized Peano continua. In order to characterize generalized continua with arbitrary end spaces we shall use trees. By a 326 R. AYALA, M. J. CHA ÂVEZ and A. QUINTERO ARCH. MATH. tree we mean a locally finite contractible graph Twith a root vertex v0such that for any vertex v jv0the number of edges containing v(the valence of v) is ^2. The root vertex induces the following ordering on the set of vertices T07T. One writes v%wif vis contained in the unique arc gwfrom wto v0. Moreover we define the height of v,jvj, as the number of vertices in the arc gv. Let Sndenote the n-th level of T; that is Sn fv2T0;jvjng. Clearly if Tn7Tis the subtree generated by the vertices with jvj%nwe have that fTngdefines an exhausting sequence in T. Finally for any v2T0let Tvdenote the subtree of Tgenerated by all w^v. Theorem 2.1. Let X be a generalized Peano continuum and T be a tree. Given any continuous map g:fX ÿ! fTthere exists a perfect map f :Xÿ!T such that fg. Moreover if gis onto f can be chosen to be a surjection. Proof. Let fKngbe an exhausting sequence in X. For each level Sjwe consider the closed and open cover Gj fgÿ1fTv;v2Sjgof fX. This cover can be refined by a cover ffUg where the sets Uare the components of XÿKnjfor some nj. We can assume n1<n2<... . Let mjnj1 and Aj UFr Kmj\U. Notice that Knj7int Kmjimplies that all Aj Uare non-empty compact sets. We choose the vertex fAj U  vj Uvif gfU 7Tv. Notice that vj U%vj1 Wif W7U. For each component U7XÿKnjÿ1we consider the finite tree TUjÿ1;jgenerated by the vertex fAjÿ1 Utogether with all vertices fAj Wwith W7U. Let DUjÿi;j7Xdenote the intersection U\KmjÿKmjÿ1. Since each finite tree is a retract of the unit square we use the Tietze extension theorem to extend f:Ajÿ1 U[ fAj W;W7Ugÿ!TUjÿ1;jto a continuous map fjÿ1 U:DUjÿ1;jÿ!TUjÿ1;j. The maps fjÿ1 Uyield a continuous map fjÿ1:KmjÿKmjÿ1ÿ!TjÿTjÿ1j^2where Tjis the tree generated by all vertices vwith jvj%j. For j0 let f0:Km1ÿ!T1be any extension of f1jFrKm1and f0x0  v0for any element x02int Km1. Then the maps fjÿ1j^1define a map f [ fjÿ1:Xÿ!T. Moreover the map fis perfect since fKmj7Tjfor j^1. Furthermore the connectedness of Ximplies that the image fXcoincides with the subtree of Tgenerated by the root vertex v0and all vertices fAj Uwith j^1 and U7XÿKnj. Therefore fX  Tif gis onto. The equality fgis straightforwardly checked from the definition of f.h In particular we have Corollary 2.2. Let X be a generalized Peano continuum with fXthe middle-third Cantor set. Then for any tree T there exists a perfect surjection f :Xÿ!T. Proof. The result follows from Theorem 2.1 since any compact metric space is the continuous image of the middle-third Cantor set ([7]; 2.5.15). h Example 2. 3. We shall use later the binary Cantor tree TCas example of tree with the Cantor set as set of ends. The tree TChas a root vertex of valence 2 and the other vertices have valence 3. The tree TChas the following canonical embedding in the unit square I2 0;10;1. 327 Vol. 71, 1998 A Hahn-Mazurkiewicz Theorem for generalized Peano continua Fig.1. The binary Cantor tree. Here the vertices at level nÿ1 nn^1correspond to the middle points of the 2nÿ1 intervals which are removed in the n-th step of the construction of the middle-third Cantor set C70;1. Now we are ready to prove the analogue of the Hahn-Mazurkiewicz Theorem for generalized continua. Theorem 2.4. Let X be a topological space. Then the following conditions are equivalent. a) X is a generalized Peano continuum. b) There exists a tree T and a perfect surjection g:Tÿ!X. Moreover T and gcan be chosen in such a way that g:fT ÿ! fXis a homeomorphism. c) There exists a perfect surjection f :TCÿ! X. d) X can be covered by an increasing sequence of Peano subcontinua fXngn^1with Xn7intXn1. In the proof of Theorem 2.4 we shall use the following relative uniform local arcwise connectedness (r.u.l.a.c.) property. Lemma 2.5. Let X be a generalized Peano continuum and K 7U7X with K compact and U open. Given " > 0there exists d>0such that if x;y2K and dx;y<dthen x and y can be joined by an arc in U of diameter smaller than ". Proof. It is similar to ([7]; 4.2.6) by using the obvious relative version of Lebesgue Lemma. h Proof of Theorem 2.4. Let fKngn^1be an exhausting sequence in X. a) )b) We construct the tree Tlevelwise as follows. The level S0consists of one vertex S0 fv0g. The level Snn^1consists of one vertex for each component of XÿKn. Moreover the vertices v2Snand w2Sn1n^0are joined by an edge hv;wiif the corresponding components Dv7XÿKnand Dw7XÿKn1verify Dw7Dv.IfT0denotes the set of vertices of T, we define a map f0:T0ÿ!Xas follows. Given v2Snits corresponding component Dvis unbounded and so Dv\Fr Kn1 j;. Then choose f0vto be a point of Dv\FrKn1. Now the construction of the perfect map g:Tÿ!Xis a variation of the proof of the classical Hahn-Mazurkiewicz Theorem (see ([7]; 4.2.7)). Firstly, we know that Xis arcwise 328 R. AYALA, M. J. CHA ÂVEZ and A. QUINTERO ARCH. MATH. connected by ([7]; 4.2.5). Moreover, the r.u.l.a.c. property holds for EnKn2ÿint Knand Bnint Kn3ÿKnÿ1. Secondly, if Chv;widenotes the copy of the Cantor middle-third set C70;1  hv;wi, we find a continuous surjection fhv;wi:Chv;wiÿ!Dv\En(see ([7]; 4.1.6)). Moreover, since Chv;wi is an homogeneous space ([8]; 30A) and it is the disjoint union of two copies of itself we can also assume that fhv;wiv  f0v 2 Fr Kn1and fhv;wiw  f0w 2 Fr Kn2. Thirdly, we extend fhv;wito a continuous map ghv;wi:hv;wi ÿ! Xwith Im ghv;wi7Bn. To do that one simply follows the proof of ([7]; 4.2.7) and uses the r.u.l.a.c. property above instead of the usual uniform local arcwise connectedness property. Let g:Tÿ!Xbe the map defined by gjhv;wighv;wi. It is easily checked that gis proper and g:fT ÿ! fXactually is a homeomorphism. b) )c) This is an immediate consequence of Corollary 2.2. c) )d) Let YnfTnwhere Tn7TCis the subtree generated by the vertices in levels %n. Then each Tnis obviously a Peano continuum and the Hahn-Mazurkiewicz Theorem implies that Ynis also a Peano continuum. Moreover, since fpreserves the local compactness ([3]; 3.7.21) Xis locally compact, and hence for each Ynwe can find an open set Gnwith Gncompact and Yn7Gn. Since fis perfect there exists knwith Tn7fÿ1Gn7fÿ1Gn7Tkn. Then Yn7Gn7Yknand so Yn7int Ykn. Now it is clear that one can inductively define an increasing subsequence Yk17Yk27... 7with Yki7int Yki1and X[ 1 i1Yki. We now take XiYkifor all i^1. d) )a) Clearly Xis a connected, locally connected and locally compact Hausdorff space. Furthermore Xis a regular space ([3]; 3.3.1) and second countable since it is a countable union of interiors of compact metric subspaces. Hence Xis metrizable by the Urysohn metrization theorem ([3]; 4.2.9), and so Xis a generalized Peano continuum. h Next we use Theorem 2.1 and 2.4 to prove the following result (compare [6]; 8.19). Corollary 2.6. Let X and Y be generalized Peano continua and g:fX ÿ! fYa continuous map. Then there exists a perfect map f :Xÿ!Y such that fg. Moreover if gis onto f can be chosen to be onto. Proof. By Theorem 2.4(b) we find a tree Tand an onto perfect map h:Tÿ!Ywith h:fT ÿ! fYa homeomorphism. By Theorem 2.1 we can find a perfect map f0:Xÿ!Twith f0 hÿ1 g:fX ÿ! fT. Then fhf0:Xÿ!Tis a perfect map with fg. Moreover if gis onto then f0and hence fare onto. h Corollary 2.7. Let X be a generalized Peano continuum and Y a Hausdorff space which is either first countable or locally compact. Then Y is the quotient space of a usc decomposition of X if and only if Y is a generalized Peano continuum and there exists an onto map g:fX ÿ! fY. We recall that a partition of X,g, is called an upper semicontinuous (usc) decomposition if each A2gis compact in Xand for each open set U7Xwith A7Uthere exists another open set V7Xcontaining Asuch that any A02gintersecting Vis contained in U. Now 2.7 follows from 2.6, 1.1 and the results on usc decomposition in Chapter I §3 of [2]. 329 Vol. 71, 1998 A Hahn-Mazurkiewicz Theorem for generalized Peano continua Acknowledgement. This work was partially supported by the project DGICYT PB961374. References [1] N. BOURBAKI, Elements of Mathematics. General Topology. Part I. Paris 1966. [2] R. J. DAVERMAN, Decomposition of manifolds. Pure Appl. Math. 126 (1986). [3] R. ENGELKING, General Topology. Sigma Ser. Pure Math. 6(1989). [4] H. FREUDENTHAL, Über die topologischen Räume und Gruppen. Math. Z. 33, 692 ± 713 (1931). [5] S. MAZURKIEWICZ, Sur les lignes de Jordan. Fund. Math. 1, 166 ± 209 (1920). [6] S. B. NADLER JR., Continuum Theory. An Introduction. Pure Appl. Math. 158 (1992). [7] A. W. SCHURLE, Topics in Topology. North-Holland 1979. [8] S. WILLARD, General Topology. Addison-Wesley Ser. Math. (1970). Eingegangen am 29. 4. 1996 Anschriften der Autoren: R. Ayala, A. Quintero Departamento de Geometría y Topología Facultad de Matema Âticas Universidad de Sevilla Apartado 1160 41080-Sevilla Spain M. J. Cha Âvez Departamento de Matema Âtica Aplicada I Escuela Universitaria de Arquitectura Te Âcnica Universidad de Sevilla Avda. Reina Mercedes s/n 41012-Sevilla Spain 330 R. AYALA, M. J. CHA ÂVEZ and A. QUINTERO ARCH. MATH.