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.