Y between continua X and Y is said to be atomic at a subcontinuumKof thedomainXprovided thatf(K) isnondegenerate andK f-I(f(K

Internat.

VOL. 21 NO. 4 (1998) 729-734

729

ATOMICITY

OF

MAPPINGS

MathematicalInstitute University of

Wroctaw

Pl. Gnmwaldzki2/4,50-384

Wroctaw,

POLAND

Mathematical Institute University of Wrodaw P1.Gnnwaldzki2/4,50-384

Wrochw, POLAND

JANUSZ J. CHARATONIK

InstitutodeMatem/ticas UniversidadNacionalAut6nomade Mxico

CircuitoExterior,CiudadUniversitaria 04510Mtxico,

D.F.,

MEXICO and

Wt.ODZIMIERZ J. CHARATONIK

Departamento

de Matemticas Facultadde Ciencias

Universidad NacionalAut6noma deMtxico CircuitoExterior, Ciudad Universitaria

04510Mtxico,

D.F.,

MEXICO

(Received December 16, 1996and in revisedform December 11,1997)

ABSTRACT. A mapping f:X--, Y between continua X and Y is said to be atomic at a subcontinuum

K

of thedomain

X

provided that

f(K)

isnondegenerate and

K f-I(f(K)).

^Theset

ofsubcontinua at which agiven mappingis atomic, considered asasubspaceof thehyperspace of all subcontinuaof

X,

isstudied. The introduced conceptisappliedtogetnew characterizationsofatomic and monotonemappings. Somerelated questionsareasked.

KEY WORDS AND PHRASES: Atomic, composition factor property, continuum, contratomic, hyperspace, mapping,monotone.

1991AMSSUBJECTCLASSIFICATION CODES: Primary 54E40,54F15. Secondary54B15.

INTRODUCTION

All spaces considered inthe paperareassumedtobe metric, and a mapping means a continuous function. Acontinuum meansacompact connectedspace. Recall thatamapping

f ^{X Y}

^betweea

continua

X

andYissaidtobemonotoneifthe inverse image of eachpoifitof

Y

(equivalently, ofeach subcontinuumof

Y)

isconnected. Asurjective mapping

f ^{X Y}

between continua

X

andYis said to be atomic provided that, for each subcontinuum

K

of

X

such that $(K) is nondegenerate,

K f-l(f(K)).

^Thenotionofan atomicmappingwasintroducedby R.D.Andersonin

[1]

^todescribe special decompositions of continua.

Soon,

atomic mappings turned out to be important tools in continuumtheoryand provedtobe interesting bythemselves, and several oftheirpropertieshavebeen discovered, e.g.in[3], [5]^and[6]. The following facton atomicmappingsisknown(see [3,Theorem 1, p.49]and[6,(4.14),p.

17]).

Fact. Everyatomicmapping ofa continuumis(hereditarily)monotone.

Thepaperconsistsoftwoparts. Inthefirst onethecompositionfactor propertyisdiscussed for the class ofatomicmappings. The second part dealswiththefamilyofsubcontinuaof the domain continuum

X

atwhichagiven mapping

f ^{X Y}

^{isatomic. In}particular,atomicmappingsaswell asmonotone ones arecharacterizedby conditions concerning thestructureofthisfamily. The paperissuppliedwith a number ofexamples;open problemsposedinboth parts of the paperindicate some directionsofafurther studyinthe area

730 J.J. CHARATONIK AND W. J. CHARATONIK

Thefollowing standard notation willbe used.

N, R

andC standfor thesets of positive imegers, reals,andcomplex numbers, respectively, equippedwith theirnatural topologies,ifneeded. Intheplane

R

thesymbol

(x, /)

meansapoint havingxand_/asits Cartesian coordinates.

1. COMPOSITION FACTOR PROPERTY

Wesaythat aclass Jdof mappings has thecomposition

factor

^propertyifthe composition g^ohof mappingshand g is in ,Monlyifg

T. Makowiakaskedin[6, (5.22),p.33]iftheclassofatomicmappingshas thecomposition factor property, and conjecturedthat itdoes.

Later,

in [7, Chapter l, Example, p. 7] he has answered his questionin thenegative. Another answerwasgiven byE.E. GraceandE. J.Voughtin[4,Section 4,p 140],who have shown that for the naturalprojection

f

of the circle ofpseudo-arcs

X

ontothe circle

Z

(which is clearly an atomic mapping) there exist a continuum

Y

and two mappings^h:X--,

Y

and

g:Y--, Z,

such that

f

^{can be factored as} the composition g^oh and g is not atomic. Both the conjecture ofMakowiakanditsnegative solution by himself andby GraceandVoughtshow that the composition factor property for the class oftheatomicmappings should bestudied in a more detailed way,andthat there are interesting problems aroundthispropertyworthwhileclarif3dng.

Ingeneral,thefollowingproblemcanbeposed.

Problem 1.1. Let

X, Y

and

Z

be continua, and let ^h:X--,

Y

and

g:Y- Z

be surjective mappings. Determine conditionsconcerning

(a)

thecontinuum

X, (b)

the continuum

Y, (c)

the mapping h, underwhichthe implication holds

if g^oh isatomic, then g is atomic (1 2)

Tobe more precise,imroduce thefollowingdefinition.

Definition 1.3.

A

class of

C

ofcontinua is said tohave thecomposition

factor

^property

for

^aclass

.M of

mappings provided that for eachcontinuum

X

ECif the composition g^ohdefinedon

X

isin

,M,

then g is in

Acontinuumis said tobedecomposableif it istheunionoftwoitspropersubcontinua. Otherwise it is said to be indecomposable. A continuum is said to be hereditarily decomposable (herechtarily indecomposable)providedthateach ofitsnondegneratesubcominua isdecomposable(indecomposable, respectively). Finally recall thataspace

X

issaidtobehomogeneousprovided that for everytwopoints pandqof

X

thereis ahomeomorphism

f ^X

^---,

^X

^{such thatf(p) q.}

Itis shownin

[2]

^{that thecircleof}pseudo-arcs(thathasbeenusedin

[4]

^asmentionedabove) is constructedinthe Euclideanplane,isdecomposable,andishomogeneous. Therefore the result ofGrace andVoughtcanbeformulatedeven in astrongerform,asfollows.

Theorem 1.4. The following classes ofcontinuadonothave the composition factorpropertyfor the class ofatomicmappings: planecontinua, decomposable continua, homogeneous cominua,aswell as the intersectionofanyof these classes.

Onthe otherhand,it isknownthat each atomic mappingdefined on an arcwiseconnectedcontinuum isahomeomorphism provided that the imagecontinuum isnondegenerate(see [6,

(6.3),

p.

51]).

Since the class of homeomorphisms obviously has the composition factor property [6, (5.14), p. 32], the following resultis immediate.

Statement1.5. Theclass ofarcwiseconnectedcontinua

(of

locallyconnectedones,inparticular) has thecomposition factor property for the class ofatomicmappings.

Thusthefollowing problemisnatural.

Problem1.6. Determinethe classes ofcontinua whichhave the composition factor property for the classofatomic mappings.

Twoparticular questions related^tothisproblemareofaspecialinterest.

731 Question 1.7. Does the class of hereditarilydecomposable continua havethe composition factor propertyfor theclassof atomic mappings?

Question 1.8. Doestheclass of hereditarily indecomposablecontinuahavethe composition factor property forthe classofatomicmappings?

Asurjective mapping h

X

^---}YbetweencontinuaXandYis saidtobeweakly

confluent

^provided

thatforeach subcontinuumQof

Y

^thereis a subcontinuumCof

X

suchthat

h(C)

^{Q. In}connection withProblem 1.1, part(c),^recallthefollowing result(see [6, (5.29),p.35]).

Proposition 1.9. If the mapping h

X Y

isweakly confluent,then implication(1.2)issatisfied.

Notethatthe conversetoProposition1.9is not true. Namelywehavethe following example Example1.10. Therearemappings h

X

^--}Yandg Y

-

^Zsuch that thecompositiong^ohand the secondmappingg areatomic,whilehis notweakly confluent.

Proof. TakeasXthe wellknown

sin(1/x)-curve

^Sdefinedby

S

{(0,y) e R2:

y

e [- 1,1]}

^U

{(x, sin(1/x)) e R2:x e (0,1]}, (1.11)

and let

L

be thelimit segment ofS. Identify thetwoend points of

L

and denotebyh

X Y

the identification mapping. ThusYisthe unionofahalflineand thecircle

h(L).

Nowletusshrink

h(L)

^to

apdint,andlet g Y

-

^Zbethequotient mapping. Thus

Z

is anarc,bothg and g^ohareatomic,while h is notweakly confluent.

Asurjective mapping h

X Y

betweencontinua

X

and

Y

is saidtobe

confluent

provided that for eachsubcontinuumQof

Y

and foreverycomponentCof theinverseimage

h-(Q)

wehave h

(C)

Q.

Since a continuum

Y

is hereditarily indecomposableifandonlyifeach mapping froma continuumontoY isconfluent(compare [6, (6.11),p.53]),^{and sinceeach}confluent mapping obviouslyisweaklyconfluent, wegetacorollarytoProposition 1.9,which isrelatedtopart(b)of Problem1.1.

Corollary1.12. If thecontinuum

Y

ishereditarily indecomposable, then implication(1.2)is satisfied.

2. ATOMICITY

Given a continuum X with a metric d, we let2x denotethe hyperspace ofallnonempty closed subsets of

X

equippedwiththe Hausdorffmetric

H

definedby

H(A,B) max{sup{d(a,B)

^aE

A},sup{d(b,A)

b

B}}

(equivalently: withtheVietoristopology,see e.g. [8, (0.1),p. and(0.12),p.

10].

Further,wedenote by

C(X)

the hyperspace of allsubcontinuaof

X,

i.e., of allconnected elementsof2

x,

and by

F1 (X)

^the

hyperspace of singletons. The readerisreferredtoNadler’s book[8] ^for,needed information on the structureofhyperspaces. Inparticular, the following is well known

(see

[8,Theorem(1.13), p.

65]).

Fact2.1. Foreach continuum

X

thehyperspace

C(X)

^isa subcontinuumof the hyperspace2

x.

Given amapping

f ^X

^-}

^Y

^{betweencontinua}

^X

^and

^Y,

we considermappings(calledthe induced

ones)

2f:

2x ₂Y and

C(f): C(X)

---*

C(Y)

definedby

2^f

(A) f(A)

^forevery

A e

2x and

C(f)(A) f(A)

^foreveryA

e C(X).

Thus,byFact2.1, thefollowingisobvious.

Fact2.2. Foreverycontinua

X

and

Y

and for each mapping

f ^X - ^Y

^wehave

^{2IIC(X) C(f)}

A proofof thenextfactisstraightfoward.

Fact 2.3. Let a mapping f:X---, Y between continua X and

Y

be given Then

C(f)(F(X))

C

FI(Y).

732 J.J. CHARATONIK AND W. J. CHARATONIK

Foranarbitrary surjection

f" ^X

^-,

^Y

^betweencontinua we considersubcominua of

X

atwhichthe mapping satisfies the atomicity condition. More precisely, given a surjective mapping

f" X Y

betweencontinua

X

and

Y,

wedenote by^.,4

(X, f)

the family of all subcontinuaKofXsuchthat

f(K)

isnondegenerateand the equality

K f-1 (f(K))

holds, i.e.,

t(X,f) {K C(X)\(C(f))-(FI(Y)) K f-(f(K))}.

(2 4) Thus thefollowing resultis aconsequence of thisdefinition.

Statement2.5. Letamapping

X Y

betweencontinua

X

and

Y

be given. Then

X

4(X, f),

so

,4(X, f)

isnonempty; (2 6)

t(X, f) c

(2.7)

Further,weput

B(X, f) A(X, /) (C(/)) - ^{(F (Y)). (28)}

Proposition 2.9. For every decreasing sequenceofcontinuabelonging

to/3(X, f)

the limit of the sequence alsois in

B(X, f).

Proof. Foreachn Nassume

K, B(X, f)

and

K+

^C

Kn.

^Put

K

Lim

K

^{and notethat}

K

f{K,.,’n N}.

Considertwocases. First, if

K,., (C(f))-I(F1 (Y))

for almost alln

N,

then

K (C(f))-I(F1 (Y)),

too, because

Fi(Y)

^is compact, and so is its preimage under

C(.f).

Thus

K B(X, f).

Second, if

K A(X, f)

^{for almostall n}

N,

then

f-l(.f(K,)) K,

for theseindices n, and wehave

f-(f(g)) f-l(f(Limg,.,)) f-l(Lim

f(g,))

f-l(N{f(g,.,)

ⁿ

N})

N{S-(f(K.)) , _{e r} N(K. e }}

g.

Thus either g

A(X,f) (if f(K)

is not a singleton), or

K (C(f))-I(F(Y)) (if

f(g) is degenerate). Consequently,

K B(X, f)

by

(2.8).

The proofis then complete.

Theexamplebelow shows that theconclusionof Proposition 2.9is not treefor arbitrary sequences of continua

In

particular, the assumption "decreasing" cannotbereplaced by"increasing" in Proposition 2.9.

Example 2.10. There is a continuum

X,

anincreasing sequence ofsubcontinua

K,,

in

X

and a monotonemapping

f"

^X

^Y

^{such that}

^{K, B(X, f)}

^{for eachn}

^N,

^whil

9

^Lim

K, B(X, f)

Proof. Let

S

be the

sin(1/x)-curve

^definexi by

(1.11).

^Put

A {(0,y) R

"y

[1,2]},

and define

X S

t3

A.

Let

Y=[O, 1],

^{and let}

f ^{X Y}

^{be theprojectiondefinedby}

f ^(x,

^{y) x. Foreach}

n N let

K f-([1/(n + 1), 1]).

^Then

^K, .A(X, f)

^C

B(X, f),

and

LimK

^{S. Since}

f(S) Y,

we have

f-(f(S))= X,

and thus LimK,

B(X,f),

as claimed. The argument is complete.

Theorem 2.11. Foreach surjective mapping

f" ^X

^--,

^Y

^{betweencontinuaXandYthefollowing}

assertionsareequivalent:

/

isatomic; (2 12)

(X, f) C(X)\(C(f))-’(F (Y));

(2.13)

B(X, f) C(X);

(2 14)

A(X, f)

is anopensubsetof

C(X).

(2 15)

73 3

Proof. Equivalence of

(2.12)

^and

(2.13)

^{is evident}from the definitions, and

(2.14)

is anotherform of (2 13)by(2.8). Thus(2.12), (2.13)and(2.14)areequivalent. The implication from(2.13)to(2 15)is obvious. Wewillshowthat

(2.15)

implies

(2.14).

Tothis aimrecall that an order arc inthehyperspace

U(X)

is afamily ofsubcontinuaofXsuchthatfor everytwomembers

A

and

B

of wehaveeither AC BorBC A. Letp be anarbitrary point ofXandlet denote an order arc in

C(X)

^from{p}to

X. By (2.15)theintersection

.A(X, f)

^{is an}opensubsetofE. DenotebyCthe componentofthe imersection that comains

X,

and by K the (only) element ofthe boundary of C in E. Then K

E

f

A(X, f).

Takeadecreasingsequence

{K,

EC rE

N}

^ofsubcontinuaofXconvergingto

K. Then K

{K

ⁿE

N),

^and since

K

ECC

.A(X, f) C/3(X, f)

by (2 8), we infer from Proposition 2.9 that

K B(X,f).

Since

K A(X,f),

we have

K (C(f))-:(F(Y))

by (2.8), whence itfollowsthat

f(K)

E

F (Y),

andtherefore,bythe definitionofanorderarc, thesubarc of from

{p)

toKis contained in

(C(f))-(F:(Y)),

^thusin

B(X, f),

whiletherestof the orderarc

,

i.e.,

C,

is contained in

.A(X, f)

byits definition.

So,

we conclude by

(2.8)

thatthe whole orderarc is comainedin

B(X, f). Now,

sincep waschosen asanarbitrary point of

X,

weinferfrom C

B(X, f)

that

C(X)

^C

B(X, f),

^whence

(2.14)

follows. Theproofiscomplete.

Thenexttheorem is a characterizationofmonotonemappingsinthe introduced terms

Theorem2.16. Foreachsurjective mapping

f:X

^{YbetweencominuaXandYthefollowing}

assertions areequivalent:

f

is monotone; (2 17)

c(f)((x, f)) c(r).

(2.1s)

Proof. Assume

f

is monotone. Since oneinclusionof equality

(2.18)

is obvious,wehavetoshow the other one. Let

L

be a nondegenerate subcominuum of

Y,

i.e. L

_ ^U(Y)\F(Y)

^Putting

K

f-l(L)

^weseethatKis a continuumby monotoneity off,and we have

-](f(K))= f-](L)

^K

ThusKE

.A(X, f)

C

13(X, f)

by

(2.4)

^and(2.8),whence

L U(f)(13(X, f)),

and(2.18)follows Assume equality

(2.18)

^holds. Takea poim ₀E

Y

and, to show that

f

is monotone, i.e., that

f-l(t0)

is connected, consider for each positive integer rz the component

L,

containing ₀ of a closed 1/r-neighborhood about _{F0 in} Y. Thus

L

^{is a} nondegenerate subcontinuum of

Y,

e,

L U(Y)\F(Y).

Againby

(2.18)

we infer that

L, C(f)(.A(X,f)),

whence

(for

^each n)there existsanondegeneratesubcontinuum

K,

ofXsuch that

f (K,)

^L,,andK,

f- (f(K,)) f- (L,)

Observe that for each n we have

L,+

C

L,,,

i.e., that the sequence

{L.,}

^is decreasing, and that

{/0} t’{L,

ⁿE

N}.

^Thusitfollows fromthe equality

K, f-(L)

^{that the}sequence

{K}

^is

decreasing,too, and wehave

Thus

f- ^(/0)

^isacominuum astheintersectionofadecreasingsequenceofcontinua

K,

Theproof isthencomplete.

Theorems2.11and (indirectly)2.16 motivatethe following question.

Question 2.19. What is the Borel class of the set

A(X,f)

considered as a subspace ofthe hyperspace

U(X)?

In connection with

(2.6)

^{of Statement 2.5} observe that the mapping h of the unit circle S

{z

EC

[z] 1}

^{(where Cstandsfor the}complex plane)

omo

itself definedby

h(z)

^z has the property that the whole S is theonlyelement of

.A(X,f),

^i.e.,

.A(X,f) {S}.

Generalizingthis phenomenon,consider thefollowing class of mappings.

Definition 2.20. Asurjectiv mapping

f ^{X Y}

between continua

X

and

Y

iscalledontratomic provided that

.A(X, f) {X).

^{Inotherwords,} a nonconstant mapping

f

^is contratomic ifthe only

734 J.CHARATONIK AND W. J. CHARATONIK

subcontinuum

K

ofX having nondegenerate image and satisfying theequalityK

f-l(f(K))

isX itself.

The above consideredmapping^hof

S

ontoitselfis anexample ofa contratomicmapping. Notethat the classof contratomic mappings doesnot containhomeomorphisms(moreover,any homeomorphism, being atomic,is notcontratomic). Inconnectionwith thisobservethat ifagainh S ^--,S is definedby

h(z)=z

^{2 and g:S1--,S} is the identity, then the compositions hog=h and goh=h are contratomic,while g isnot. Thisleadstothefollowingobservation.

Observation 2.21. The class ofcontratomic mappings does not have the composition factor property.

Proposition 2.22. Leth

X Y

and g:

Y

^---,

Z

be surjcctive mappings between continua

X, Y

and

Z,

respectively. If either horgiscontratomic, then the composition g^ohiscontratomic,too

Proof. Forany

K

CXwehave

K

C

h-(h(K))

and

h(K)C g-(g(h(K)))

Let

K _ ^C(X)\{X}.

^{If h is} contratomic, then

Kh-(h(K));

and if g is contratomic, then h(g)

g-(g(h(g))),

whichleadsto

g_C

h-(h(g))h-(g-(g(h(g))))

sincehissurjective. Putting

f

^goh,weget

K h-(g-(g(h(K)))) f-(f(K))

^{in eitherofthe}

twoconsidered cases The proofis thencomplete.

REFERENCES

ANDERSON,

R.D.,

Atomicdecompositions of continua,DukeMath.

J.

24(1956),^507-514

[2] BING,

R.H. and

JONES, FB.,

Anotherhomogeneous planecontinuum, Trans. Amer.Math. Soc.

90(1959),171-192.

[3] EMERYK, A.andHORBANOWICZ,

Z.,

Onatomicmappings,Colloq.Math. 27(1973),49-55.

[4] GRACE, E.E. and VOUGHT,

E.J.,

Four mapping problems of Madkowiak, Colloq. Math. 69 (1995),133-141.

[5] HOSOKAWA,

H.,

Some remarks on the atomic mappings, Bull. Tokyo Gakugei Univ. (4), ⁴⁰ (1988),31-37.

[6] MAtKOWIAK, T.,

Continuousmappingsof continua,DissertationesMath. (RozprawyMat.)158 (1979),1-91.

[7] MAKOWIAK, T.,

Singulararc-likecontinua,DissertattonesMath. (Rozprawy

Mat.)

²⁵⁷

(1986),

1-35.

[8] NADLER,

S.B.,

Jr., Hyperspaces ofSets,

M. Dekker, 1978.