A Hahn-Mazurkiewicz Theorem for generalized Peano continua

A Hahn-Mazu kiewicz Theo em o gene alized Peano con inua By R. AYALA,M.J.CHA ÂVEZ and A. QUINTERO Abs ac . S. Mazu kiewicz ema ked a e his p oo o he celeb a ed Hahn- Mazu kiewicz Theo em ([5]) ha any gene alized Peano con inuum is he con inuous image o he hal -line 0;1 bu he con e se does no hold. The e o e con inuous images o 0;1do no cha ac e ize gene alized Peano con inua. In his pape we use pe ec maps and ees o ob ain an analogue o he Hahn-Mazu kiewicz Theo em o gene alized Peano con inua in he spi i o he classical Hahn-Mazu kiewicz Theo em. 0. In oduc ion. The well known Hahn-Mazu kiewicz Theo em ([5]) es ablishes ha a Hausdo space Pis a Peano con inuum i and only i i is he image o he uni in e al I0;1. This heo em was he culmina ion o he s udy s a ed by Peanos celeb a ed example o a squa e- illing cu e. When compac ness is eplaced by local compac ness we s ill ha e an in e es ing class o opological spaces, namely, gene alized Peano con inua. These spaces we e al eady conside ed by he ounde s o con inuum heo y. In ac , Mazu kiewicz in his seminal pape [5] showed ha any gene alized Peano con inuum is he con inuous image o he hal - line 0;1 bu he con e se does no hold. The e o e con inuous images o 0;1 do no cha ac e ize gene alized Peano con inua. In his pape we use pe ec maps and ees o ob ain an analogue o he Hahn-Mazu kiewicz Theo em o gene alized Peano con inua (Theo em 2.4). In spi e o i s simple na u e his esul seems o be new in he li e a u e. We use his heo em o gi e wo u he esul s on gene alized con inua which ex end wo basic esul s on Peano con inua (Co olla ies 2.6 and 2.7). 1. Gene alized Peano con inua. In his pape we shall deal wi h he class o gene alized Peano con inua. We ecall ha a con inuum Xis a compac connec ed me izable space. When compac ness is eplaced by local compac ness he space Xis called a gene alized con inuum. I in addi ion Xis locally connec ed i is called a (gene alized) Peano con inuum. Hence, any (gene alized) Peano con inuum is a cwise connec ed by ([7]; 4.2.5). Mo eo e i ollows om ([3]; 4.4 F.(c)) ha any gene alized con inuum is sepa able and hence second coun able and s-compac ( [3]; 4.1.16, and 3.8.c(b) ). The local compac ness oge he wi h he s-compac ness yield ha Xis a coun able union [ 1 n1Kno compac subse s Kn7Xwi h Kn7in Kn1. Ac ually we can assume wi hou loss o gene ali y ha each Knis connec ed and all he componen s o XÿKna e unbounded. Indeed, each Knis con ained in a ini e Ma hema ics Subjec Classi ica ion (1991): 54F15, 54E40. union o open connec ed subse s o compac closu e K0 n.I K0 nis no connec ed we can conside a new K0 nby adding o K0 ncompac and connec ed subspaces joining i s ( ini e) componen s. I some componen s o XÿKna e bounded hen we conside a new K00 nby adding o Knall he bounded componen s in XÿKn. The sequence Kngn^1wi h hese wo p ope ies will be called an exhaus ing sequence in X. Gi en an exhaus ing sequence in X Kngn^1aF euden hal end o Xis a sequence " Cnn^1o componen s Cn7XÿKnwi h Cn17Cn. We deno e by X he se o F euden hal ends o X. The se b XX[ Xadmi s a compac opology whose basis consis s o he open se s o X oge he wi h he se s c CnCn[ "2 X;Cnappea s in "g n^1. This opology (which does no depend on he sequence Kngn^1) is called he F euden hal opology and b Xis called he F euden hal compac i ica ion o X. Mo eo e he subspace X u ns ou o be homeomo phic o a closed subse o he Can o se (see ([4]) o de ails). Ape ec map :Xÿ!Yis a con inuous closed map such ha o each y2Y he ibe ÿ1yis compac . I Xand Ya e locally compac Hausdo spaces is pe ec i and only i i is con inuous and ÿ1Kis compac o any compac subse K7Y(i.e. is a p ope map ([1]; I.10.2.1)). I Xand Ya e gene alized con inua any pe ec (o equi alen ly p ope ) map :Xÿ!Y ex ends o a con inuous map b : b Xÿ! b Ywhich es ic s o a con inuous map : X ÿ! Y. Namely i " Cnn^1; b "  "  Dkk^1whe e Cnk  Dk o some inc easing subsequence Cnkk^1o ". The classical Hahn-Mazu kiewicz Theo em es ablishes ha a Hausdo space Pis a Peano con inuum i and only i i is he con inuous image o he uni in e al 0;1. In pa icula Peano con inua a e p ese ed by con inuous maps. In addi ion o he abo e heo em S. Mazu kiewicz ([5]) also shows ha any gene alized Peano con inuum is he con inuous image o he hal -line 0;1  bu he con e se does no hold; ha is, gene alized Peano con inua a e no p ese ed by con inuous maps. In ac hey a e p ese ed by pe ec maps. Namely Lemma 1.1. Le X be a gene alized Peano con inuum and :Xÿ!Y a pe ec su jec ion. Then Y is a gene alized Peano con inuum. P oo . Indeed, pe ec maps p ese e all p ope ies which de ine Peano con inua (see [3]; 4.4.15, [3]; 3.7.21, and [7]; 3.5.7(4)). h Fu he mo e he ollowing lemma shows ha mos o gene alized Peano con inua a e no pe ec images o 0;1 . In pa icula only one-ended spaces can be pe ec images o 0;1 . Lemma 1.2. Le X and Y be a gene alized Peano con inua. Any pe ec su jec ion :Xÿ!Y induces a con inuous su jec ion b : b Xÿ! b Y and : Xÿ! Y. P oo . Since b Xand b Ya e Hausdo compac spaces b is a closed map and hence X  b Y7 b  b X. Tha is, b and a e on o maps. h 2. An analogue o he Hahn-Mazu kiewicz Theo em o gene alized Peano con inua. In o de o cha ac e ize gene alized con inua wi h a bi a y end spaces we shall use ees. By a 326 R. AYALA, M. J. CHA ÂVEZ and A. QUINTERO ARCH. MATH. ee we mean a locally ini e con ac ible g aph Twi h a oo e ex 0such ha o any e ex  j 0 he numbe o edges con aining ( he alence o ) is ^2. The oo e ex induces he ollowing o de ing on he se o e ices T07T. One w i es %wi is con ained in he unique a c gw om w o 0. Mo eo e we de ine he heigh o ,j j, as he numbe o e ices in he a c g . Le Sndeno e he n- h le el o T; ha is Sn 2T0;j jng. Clea ly i Tn7Tis he sub ee gene a ed by he e ices wi h j j%nwe ha e ha Tngde ines an exhaus ing sequence in T. Finally o any 2T0le T deno e he sub ee o Tgene a ed by all w^ . Theo em 2.1. Le X be a gene alized Peano con inuum and T be a ee. Gi en any con inuous map g: X ÿ! T he e exis s a pe ec map :Xÿ!T such ha g. Mo eo e i gis on o can be chosen o be a su jec ion. P oo . Le Kngbe an exhaus ing sequence in X. Fo each le el Sjwe conside he closed and open co e Gj gÿ1 T ; 2Sjgo X. This co e can be e ined by a co e Ug whe e he se s Ua e he componen s o XÿKnj o some nj. We can assume n1<n2<... . Le mjnj1 and Aj UF Kmj U. No ice ha Knj7in Kmjimplies ha all Aj Ua e non-emp y compac se s. We choose he e ex Aj U  j U i g U 7T . No ice ha j U% j1 Wi W7U. Fo each componen U7XÿKnjÿ1we conside he ini e ee TUjÿ1;jgene a ed by he e ex Ajÿ1 U oge he wi h all e ices Aj Wwi h W7U. Le DUjÿi;j7Xdeno e he in e sec ion U KmjÿKmjÿ1. Since each ini e ee is a e ac o he uni squa e we use he Tie ze ex ension heo em o ex end :Ajÿ1 U[ Aj W;W7Ugÿ!TUjÿ1;j o a con inuous map jÿ1 U:DUjÿ1;jÿ!TUjÿ1;j. The maps jÿ1 Uyield a con inuous map jÿ1:KmjÿKmjÿ1ÿ!TjÿTjÿ1j^2whe e Tjis he ee gene a ed by all e ices wi h j j%j. Fo j0 le 0:Km1ÿ!T1be any ex ension o 1jF Km1and 0x0  0 o any elemen x02in Km1. Then he maps jÿ1j^1de ine a map  [ jÿ1:Xÿ!T. Mo eo e he map is pe ec since Kmj7Tj o j^1. Fu he mo e he connec edness o Ximplies ha he image Xcoincides wi h he sub ee o Tgene a ed by he oo e ex 0and all e ices Aj Uwi h j^1 and U7XÿKnj. The e o e X  Ti gis on o. The equali y gis s aigh o wa dly checked om he de ini ion o .h In pa icula we ha e Co olla y 2.2. Le X be a gene alized Peano con inuum wi h X he middle- hi d Can o se . Then o any ee T he e exis s a pe ec su jec ion :Xÿ!T. P oo . The esul ollows om Theo em 2.1 since any compac me ic space is he con inuous image o he middle- hi d Can o se ([7]; 2.5.15). h Example 2. 3. We shall use la e he bina y Can o ee TCas example o ee wi h he Can o se as se o ends. The ee TChas a oo e ex o alence 2 and he o he e ices ha e alence 3. The ee TChas he ollowing canonical embedding in he uni squa e I2 0;10;1. 327 Vol. 71, 1998 A Hahn-Mazu kiewicz Theo em o gene alized Peano con inua Fig.1. The bina y Can o ee. He e he e ices a le el nÿ1 nn^1co espond o he middle poin s o he 2nÿ1 in e als which a e emo ed in he n- h s ep o he cons uc ion o he middle- hi d Can o se C70;1. Now we a e eady o p o e he analogue o he Hahn-Mazu kiewicz Theo em o gene alized con inua. Theo em 2.4. Le X be a opological space. Then he ollowing condi ions a e equi alen . a) X is a gene alized Peano con inuum. b) The e exis s a ee T and a pe ec su jec ion g:Tÿ!X. Mo eo e T and gcan be chosen in such a way ha g: T ÿ! Xis a homeomo phism. c) The e exis s a pe ec su jec ion :TCÿ! X. d) X can be co e ed by an inc easing sequence o Peano subcon inua Xngn^1wi h Xn7in Xn1. In he p oo o Theo em 2.4 we shall use he ollowing ela i e uni o m local a cwise connec edness ( .u.l.a.c.) p ope y. Lemma 2.5. Le X be a gene alized Peano con inuum and K 7U7X wi h K compac and U open. Gi en " > 0 he e exis s d>0such ha i x;y2K and dx;y<d hen x and y can be joined by an a c in U o diame e smalle han ". P oo . I is simila o ([7]; 4.2.6) by using he ob ious ela i e e sion o Lebesgue Lemma. h P oo o Theo em 2.4. Le Kngn^1be an exhaus ing sequence in X. a) )b) We cons uc he ee Tle elwise as ollows. The le el S0consis s o one e ex S0 0g. The le el Snn^1consis s o one e ex o each componen o XÿKn. Mo eo e he e ices 2Snand w2Sn1n^0a e joined by an edge h ;wii he co esponding componen s D 7XÿKnand Dw7XÿKn1 e i y Dw7D .I T0deno es he se o e ices o T, we de ine a map 0:T0ÿ!Xas ollows. Gi en 2Sni s co esponding componen D is unbounded and so D F Kn1 j;. Then choose 0  o be a poin o D F Kn1. Now he cons uc ion o he pe ec map g:Tÿ!Xis a a ia ion o he p oo o he classical Hahn-Mazu kiewicz Theo em (see ([7]; 4.2.7)). Fi s ly, we know ha Xis a cwise 328 R. AYALA, M. J. CHA ÂVEZ and A. QUINTERO ARCH. MATH. connec ed by ([7]; 4.2.5). Mo eo e , he .u.l.a.c. p ope y holds o EnKn2ÿin Knand Bnin Kn3ÿKnÿ1. Secondly, i Ch ;wideno es he copy o he Can o middle- hi d se C70;1  h ;wi, we ind a con inuous su jec ion h ;wi:Ch ;wiÿ!D En(see ([7]; 4.1.6)). Mo eo e , since Ch ;wi is an homogeneous space ([8]; 30A) and i is he disjoin union o wo copies o i sel we can also assume ha h ;wi   0  2 F Kn1and h ;wiw  0w 2 F Kn2. Thi dly, we ex end h ;wi o a con inuous map gh ;wi:h ;wi ÿ! Xwi h Im gh ;wi7Bn. To do ha one simply ollows he p oo o ([7]; 4.2.7) and uses he .u.l.a.c. p ope y abo e ins ead o he usual uni o m local a cwise connec edness p ope y. Le g:Tÿ!Xbe he map de ined by gjh ;wigh ;wi. I is easily checked ha gis p ope and g: T ÿ! Xac ually is a homeomo phism. b) )c) This is an immedia e consequence o Co olla y 2.2. c) )d) Le Yn Tnwhe e Tn7TCis he sub ee gene a ed by he e ices in le els %n. Then each Tnis ob iously a Peano con inuum and he Hahn-Mazu kiewicz Theo em implies ha Ynis also a Peano con inuum. Mo eo e , since p ese es he local compac ness ([3]; 3.7.21) Xis locally compac , and hence o each Ynwe can ind an open se Gnwi h Gncompac and Yn7Gn. Since is pe ec he e exis s knwi h Tn7 ÿ1Gn7 ÿ1Gn7Tkn. Then Yn7Gn7Yknand so Yn7in Ykn. Now i is clea ha one can induc i ely de ine an inc easing subsequence Yk17Yk27... 7wi h Yki7in Yki1and X[ 1 i1Yki. We now ake XiYki o all i^1. d) )a) Clea ly Xis a connec ed, locally connec ed and locally compac Hausdo space. Fu he mo e Xis a egula space ([3]; 3.3.1) and second coun able since i is a coun able union o in e io s o compac me ic subspaces. Hence Xis me izable by he U ysohn me iza ion heo em ([3]; 4.2.9), and so Xis a gene alized Peano con inuum. h Nex we use Theo em 2.1 and 2.4 o p o e he ollowing esul (compa e [6]; 8.19). Co olla y 2.6. Le X and Y be gene alized Peano con inua and g: X ÿ! Ya con inuous map. Then he e exis s a pe ec map :Xÿ!Y such ha g. Mo eo e i gis on o can be chosen o be on o. P oo . By Theo em 2.4(b) we ind a ee Tand an on o pe ec map h:Tÿ!Ywi h h: T ÿ! Ya homeomo phism. By Theo em 2.1 we can ind a pe ec map 0:Xÿ!Twi h 0 hÿ1 g: X ÿ! T. Then h 0:Xÿ!Tis a pe ec map wi h g. Mo eo e i gis on o hen 0and hence a e on o. h Co olla y 2.7. Le X be a gene alized Peano con inuum and Y a Hausdo space which is ei he i s coun able o locally compac . Then Y is he quo ien space o a usc decomposi ion o X i and only i Y is a gene alized Peano con inuum and he e exis s an on o map g: X ÿ! Y. We ecall ha a pa i ion o X,g, is called an uppe semicon inuous (usc) decomposi ion i each A2gis compac in Xand o each open se U7Xwi h A7U he e exis s ano he open se V7Xcon aining Asuch ha any A02gin e sec ing Vis con ained in U. Now 2.7 ollows om 2.6, 1.1 and he esul s on usc decomposi ion in Chap e I §3 o [2]. 329 Vol. 71, 1998 A Hahn-Mazu kiewicz Theo em o gene alized Peano con inua Acknowledgemen . This wo k was pa ially suppo ed by he p ojec DGICYT PB96- 1374. Re e ences [1] N. BOURBAKI, Elemen s o Ma hema ics. Gene al Topology. Pa I. Pa is 1966. [2] R. J. DAVERMAN, Decomposi ion o mani olds. Pu e Appl. Ma h. 126 (1986). [3] R. ENGELKING, Gene al Topology. Sigma Se . Pu e Ma h. 6(1989). [4] H. FREUDENTHAL, Übe die opologischen Räume und G uppen. Ma h. Z. 33, 692 ± 713 (1931). [5] S. MAZURKIEWICZ, Su les lignes de Jo dan. Fund. Ma h. 1, 166 ± 209 (1920). [6] S. B. NADLER JR., Con inuum Theo y. An In oduc ion. Pu e Appl. Ma h. 158 (1992). [7] A. W. SCHURLE, Topics in Topology. No h-Holland 1979. [8] S. WILLARD, Gene al Topology. Addison-Wesley Se . Ma h. (1970). Eingegangen am 29. 4. 1996 Ansch i en de Au o en: R. Ayala, A. Quin e o Depa amen o de Geome ía y Topología Facul ad de Ma ema  icas Uni e sidad de Se illa Apa ado 1160 41080-Se illa Spain M. J. Cha  ez Depa amen o de Ma ema  ica Aplicada I Escuela Uni e si a ia de A qui ec u a Te Âcnica Uni e sidad de Se illa A da. Reina Me cedes s/n 41012-Se illa Spain 330 R. AYALA, M. J. CHA ÂVEZ and A. QUINTERO ARCH. MATH.