AND WORDS

VOL. 20 NO. 4 (1997) _681-688

LOCAL

CONNECTIVITY AND

MAPS ONTO

NON-METRIZABLE ARCS

J.NIKIEL

AmericanUniversity of Beirut Beirut, Lebanon [email protected]

and L.B. TREYBIG Texas

A & M

University CollegeStation,Texas 77843-3368,USA

treybigmath.tamu.edu and

H.M.TUNCALI NipissingUniversity

NorthBay, Ontario, Canada P1B 8L7 murat einstein.unipissing,ca

(Received December 15, 1995 and in revised form July 6, 1996)

ABSTRACT.Threeclasses of locallyconnected continua which admitsufficientlymanymaps ontonon-metricarcs areinvestigated. Itisproved that allcontinuainthose classesarecontin- uousimages ofarcsand,therefore,have other quiteniceproperties.

KEY

WORDSAND PHRASES: arc,locally connected continuum,monotonicallynormal, rim-countable, rim-finite, rim-metrizable,rim-scattered

1991AMS SUBJECT CLASSIFICATION CODES: Primary54F15, Secondary 54C05 54F05

INTRODUCTION

Let

C

denote the class of allHausdorif continuousimages of orderedcontinua.

In

the last three decades the class C has been studied extensivelyby a number of authors

(see

e.g.

[2],

[4], [6-8], [11-13], [16-22], [26]

^and

[27]).

^Two results fromthisstudy havesuggested that the investigation couldnaturally be extended tothe largerclass

TiM

^ofallrim-metrizable, locally connected continua. Namely,

(1)

ⁱⁿ

[8]

^{in 1967 Mardeid proved} that each element ofC has abasisofopen

Fa-sets

with metrizableboundaries, and

(2)

in

[4]

in 1991 Grispolakis, Nikiel, Simoneand Tymchatyn showed that ifa set

P

is irreduciblewith respect tothe property of beingacompact setwhichseparates the element

X

of

C,

then

P

ismetrizable.

Ihis 1989 thesis

[23]

^andtwosubsequentpapers

[24]

^and

[25]

Tuncalibegananinvestigation of the class

7M

and continuousimages ofelements of that class. He showed that Treybig’s product theorem of

[18]

^{which holds}in Cis nolonger valid in

RM. ^However,

^{he provedthat}

Mardeid’stheorem forConpreservationof weight by light mappingsis true in

RM, [25].

^He

also consideredthe class

T.s

^ofallrim-scattered, locally connectedcontinua, andtheclass

Rc

of alltim-countable, locallyconnected continua. Later,Nikiel, Tuncali andTymchatyn gavean exampletoshow that

R.c

^{is nota}subclass of

t?, [15].

Then,recently the authors ofthispaper showed thethe continuousimage ofanelement of

7ZM

^{need not bein}

R.M, [14].

Furthermore, DrozdovskyandFilippovprovedin

[3]

^that

Rs

^isalargerclass of spaces than/Zc.

Also, in 1973 Heath, Lutzer and

Zenor, [5],

^showed that every linearly ordered ordered topologic space and each of its Hausdorffcontinuous and closed images aremonotonically normal.

In [10]

in1986Nikielaskedifevery monotonically normal compactumisthecontinuous image ofacompact ordered space. That problem still remainsopen.

In

what follows we let

R.MN

denote the class of monotonicallynormal, locallyconnected continua. Ourfirstresult is

th,

efollowing:

THEOREM 1. If

X

E

M

^J

S

^U

.MN

andfor each pairofpoints a, b EXthereexists acontinuousontomap

f: ^X [c, d]

^{such that}

f(a)

c,

f(b)

^{d and}

[c, d]

^isanon-metrizable arc, then

X

?.

We

note that alargeclassof examples satisfying the properties of

X

above can becon- structedasfollows:

In [1]

ⁱⁿ¹⁹⁴⁵

Arens

studiedthe class oflinearhomogeneouscontinua, that isthe class ofarcswhich areorderisomorphictoeach oftheirsubarcs. Arens showed, thatup toahomeomorphism,thereexistatleast

R1

^membersof

,

includingthe real numbersinterv

[0,1].

Thus,somespaces

X

asin Theorem 1 could beobtainedby pasting togethercopies of any

Z .

Ifasubset

B

ofaspace

P

containsnodense-in-itself, non-emptysubset,wesay that

B

is scattered.

In

this paper the definitionofmonotone normality we use is anequivalent onegiven in

Lemma

2.2

(a)

of

[5]. It

says that a space

P

is monotonically normalprovided there is an operator

G

which assigns to

ear

ordered pair

(S, T)

^ofmutually separated subsets of

P

an open set

G(S, T)

^{such that}

(i)

^SC

G(S,T)

C

cl(G(S,T))

CP-

T,

and

(ii)

if

(S’,T’)

^{is alsoa pair of}mutually separatedsets such that S C

S’

and

T’

C

T,

then

G(S, T) c (S’, T’).

PROOF OF

THEOREM

1.

Suppose

that

X

is nothereditarily locally connected. Then, thereexistsasubcontinuum

C

of

X

such that Cfails tobe connectedim kleinen at thepoint p. Utilizingtheideas inTheorem 11, p. 90,ofMoore

[9],

there existsaconnectedopenset U containingp,asequenceR1, R2,R3,...of connected open in

X

^setscontainingp,andasequence

G1,G2, G3,... ofcontinuasuch that

(2)

^G.R,

#and ^{G,R.+x =@}

^{forn 1,2,3}

(3)

eachG, isacomponent ofU[3Cand

G,

⁽³

bd(U) y

forn 1, 2,3,...;and

(4) G.

^f’l

Gm= 1

ifn rn, and there existmutuallyexclusive opensetsV1, V2,

Vs,.

such that G,C

V,,

forn 1,2,3,...

Foreach positive integer n let H, be a component ofG,

R

which intersects

bd(Rl)

and

bd(U),

andlet s.

H.

^Iq

bd(R1)

^{and t, H, N}

bd(U).

^Let

H0

denote the limiting ^set of the sequence

H,

H2,H3,...; whichby definitionis theset ofall x such thateveryopen set containingx intersectsinfinitely manysetsH,.

Let

L

^(resp.

L2)denote

the limitingset of

{sl}, {s2}, {3},

(resp.

{t}, {t2}, {t3}, ...).

Thereexists

(s, t) L

^x

L2

^{so thatifVisa}neighborhood^{ofs and}Wis aneighborhoodof t, then

(s., t.)

belongsto VxWfor infinitelymanyn.

Weshallshow that some component of

H0

^{contains s,}

}.

If not, then

H0

^{is theunionof}

twomutually separatedsetsSand

T

suchthats

S

and T. Thereexistdisjoint open sets V andWsothat S CVand

T

CW. Then

(sn, t,)

belongs toV x Wfor infinitelymany^n.

Sinceeach

H.

^{isacontinuum,}

^H,

^gl

(X (V

U

W)) 1

for infinitelymanyn. It follows that somepoint of

H0

^{lies in}

X (V

U

W),

^acontradiction.

Let

f" ^X [c,d]

^{bea continuousmap ontoa} non-metrizablearc

[c,d],

^where

f(s)

^c and

f(t)

d. Thereisanincreasingsequencenl, n2,n3 of positive integers such that

(1) f(s.,) >_ f(s..+,)

^and

f(t.,) <_ f(t.,+,)

^{for 1,2,...;}

(2) f(s,.)

^cand

f(t,,)

d; and

(3) [f(s., ), f(t,,)]

^{isnotmetrizablefor 1,2}

Let

c’= f(snt)

^andd’=

f(t, t).

Our proofnowdivides intothreecases.

CASE 1.

X 7M. For

each n

>

2 let

M,

denote a metrizable closed set lying in

X- Urn--1 ^Ht

such that if1

< <

j

<

n, then

H,

and Hj ^are

selarated

ⁱⁿ

^X

^by

^{M,. Let}

D.

^{denoteacountablesetdensein}

M,

forn 2, 3,... Weintend toshow that

f 0=2 ^Dr)

^is

dense in

[c, d],

^{whichwouldmeanthat}

[c, d]

^isseparable, and therefore metric,acontradiction.

Let x(5

]c, d[

^{andlet c}

<

^u

<

x

<

^v

<

^din the natural ordering of

[c, d].

The components of

f-(]u,

^v

D

^{whichhavelimitpointsinboth}

^f-(u)

^and

^f-(v)

^{canbe labeledP,P2,...}

^,P,o.

Let

No

^{be an}integer such that if

>_ No

^then s,,

f-l([c,u[)

^and

t,, f-(]v,d]).

^There

existtwo of

No,No +

^1,...

,No +

^{no, say and j,such that}

Ha.

^and

Ha,

^{bothintersectthesame}

Pt,

which mustthenintersectsomeD,.. Therefore,

0=2 ^f(D)

^intersects

^{]u, v[.}

CASE2.

X 7MN.

^Foreaz.h 1,2,... let Qi denoteacomponent of

Hn, CIf - ^{([c’,d’])}

which intersects

f-(c’)

^and

f-(d’),

^{and let} Q0 denote the limiting set of

Q,

Q2, Q3,... We note thatsomecomponentofQ0 intersectsboth

f-(c’)

^and

f-l(d’)

^{sinceeverymap ontoan}

arcisweakly confluent.

By Remark 2.3

(c)

^of

[5], ^Z [,.J,__0 Q,,

is monotonicallynormal; so let beamonotone normality operatoron

Z

asin the earlier definition. Foreach closed set F in

[c ,

^d

]

^let

Q

F

{x f(x)

6-

F

and x 6- Z-

Q0},

and let

RF ^{x f(x)

6-

[c’,d’]-

^{F and x 6-}

Q0}.

Now,

QF and

RE

^are mutually separated subsets of

Z;

^so for each positive integer n, let

T(F,n) {y

^6_

[c’,d’]

y

f(x)

forsome x 6-

Q, Cl(QF,RF)}.

^{It canbe shown that Tis}

stratificationfor

[c’, d’].

^{Since each}stratifiablecompact space ismetrizable,

[c’, d’]

^ismetrizable, acontradiction.

CASE3.

X

6-

Ts.

^Foreach/=1,2,3,...letK,denoteacomponent of

H,. fir

^-1

([c’, d’])

which intersects

f-1 ^(c,)

^and

f-1 ^(d’).

Wehavetoconsider some subcases.

CASE 3A.

[c’, d’]

^containsuncountablymanymutuallyexclusive open sets.

CASE

3A1. [c’,d’]

^{does not}satisfy thefirst axiomof countability. Thus,withoutlossof generality,assumethat there isasubset

{da

^a

< w

of

[c’, d’]

such that

a <

a2 implies that

da, < da2

ⁱⁿ

^{[c’, d’],}

^and

^da

^d’.

Let

K0

^{denote thelimitingset}of K1, K2, K3,... Let

Q

denote acomponent of

K0

^which

intersectsboth

f-(c’)

and

f-(d’).

Foreacha

<

Wl let

Wa

^denoteaconnected open set such that

Wa

^containsapointXaofQcl

f-(]da, da+[),

^and

Wa ^c f-(]da, do,+[).

There existsapositiveinteger

no

^{and acofinal}subsequence

{dan }

^of

da

^{such that}

Kn0 # ^@

^forall a0. Foreach ₇

< wx

^let

L,

denote the closureof the set

0_>, Wa.

^Let

L -<,01 Lv.

Observe that ify6-

L,

then each openneighborhood ofy intersectsuncountably many sets

War.

^{Let Wbea}component ofL. Notethat

WClK, # ⁰ #

^QclWandWC

f-(d’).

Thus,Wis anon-degeneratecontinuum.

Let

M0

^and

M1

be connectedopen setssuchthat

00

^N

M---’" ⁰

^and

^M,

^ClW

# ⁰

^{for 0,1.}

Let

{Mo, M }.

Nowsupposethat

,

^hasbeen chosen and consistsof2 mutuallyexclusive connected open sets such that if

G,G’

q

,

^and

^G # ^G’,

^then

C

^’’7G

1

^and GCl

I # ^l # ^G’

^NW.

^For

each

G’

6-

,

^let

_G

^and

_G

^{bemutuallyexclusiveconnectedopen sets suchthat G}o Cl

G1 GUG

^{C G’ and}

GW # ⁰ # G

^{W. Let}

^,+ {F- F

G or

F G

^{for some}

G’

6-

g,}.

Foreachnlet

H: n

^{andlet H}

[n=l gn"

Thereexists$0

< w

^{such that}

^G’

^Cl

f-(do

^{for each}

^G’

^6-

U= .

^Thereexistsa

closedscattered set ^cin

X

whichseparates

f- ([c, d0]

^from

f-(d’). However,

^CglHcontains a perfect set because c Cl

H

can be mapped onto a

Cantor

set, and it is wellknown that a scatteredsetcannotbemappedcontinuouslyonto aperfect set. Thisisa contradiction.

CASE

3A2. [c’, d’]

^satisfiesthefirst axiomof countability ateach point. Let

{]ca,

a

< w }

^denoteanuncountablecollection ofmutuallyexclusive open intervals in

]c’, d’[.

Using the localconnectivityof

X

wefindthat for eachathereexistsonlyafinitenumber,sayha,of components of

f- (]ca, da[)

^whichhavelimit points in both

f-(ca)

and

f-l(d,,).

Someinteger

No n=

repeats foruncountablymany

a,’s;

sowemaysupposewithoutloss of generality that

n,,

No

^{for eacha}

<

w.

Thereexists aclosed scattered set Ssuch thatS separates

K,

from Kj for each pairi, such that 1

_< <

j

_< No +

^{1. Thus,sincefor}each a, each set

K,

where 1

_< _< No +

^has

the property that somecomponent of

K,

N

f-l(]ca, d,[)

^{has limitpoints inboth}

f-’(ca)

^and

f-l(d,),

^itfollows that Smust intersect each

f-1 (]ca, da[).

Since

[c’, d’]

^isfirst countable,thereexist collections1,2,g3,--- such that

(1)

^each

consists of 2 mutually exclusive closed intervals in

[c’,d’],

^and

(2)

each element of each containsexactly two elements of,,+1 ^{and contains}uncountably many elements of

{]ca,d[:

<wl}.

Foreachpositiveintegernlet

L’, [.J,,,

^andlet

^L’ --1L.

^{Wefindthat S}

^f-(L’)

containsaperfectset,acontradiction.

CASE 3B.

[c,d]

^is not metrizable and does not contain uncountably many mutually exclusiveopen sets

(i.e.,

^itisaSouslin

line).

^Thus,

[c’, d’]

^satisfiesthefirst axiomof countability.

If thereexists acollectionof metrizable open intervals whoseunion isdense in

[c , d],

^wefind that

[c’,

^d

]

^ismetrizable since it isseparable.

Hence,

withoutloss of generalitywemay assume that

[c , d’]

^{containsnometrizable}subinterval.

Similarlyasabove, for each

Ix, y[

C

[d, d’]

^welet

nz

denote the numberofcomponents of

f-(]z,y[)

with limit points in both

f-(x)

^and

f-(y).

CASE

3B. Suppose

^thereexistsapositive integer

No

^{and a}subinterval

Ix, y[

^of

such that ifx

_<

z

<

w

_<

y, then n,

_< No.

^{Let S} be aclosed scatteredset such that if 1

_< <

j

_< No +

^{1, thenSseparates}

K,

fromKj. Usingthe ideasfromCase

3A

^{we findthat}

ifx

_<

z

<

^w

_<

y, then S

]’-(]z, w[) # .

^Therefore,

^f(,.S)

^:3

^{Ix, y],}

which contradicts the well-known fact thatascattered compactumcannotbemappedontoaperfect set.

CASE

3B2.

^{Assumethat forevery}

Ix, y[

C

[c’, d’]

^{thereexistsaninterval}

]z, w[

C

Ix, y[

^such thatn,o

> n=.

Foreach positiveintegernlet

,,

be maximal relativetothepropertyofbeingacollection of mutuallyexclusive open intervalslyingin

[d, d’]

^suchthatif

Ix, y[ ,,

^then_nffi ^n. Notethat each

,,

isat mostcountable. Let

S,,

denote thesetof all end-points ofintervalswhichbelong to

,,. We

aregoing^toshowthat

U,,__ Sn

^isdensein

[c’, d’],

and thus obtainacontradiction.

Let

Ix, y[

C

[d, d].

^Thereexists

]z, w[

C

Ix, y[

such that

n > nz.

^Thus,x

^#

^{z ory}

^#

^w.

By

maximality of

,,,,

^thereexists

^Is, t[ e ,,,

^{such that}

^{]s, t[}

^g

^]z, w[ #

^{}. But}

Is, t[ ]x, y[,

andso s

e Ix, y[

or

e Ix, y[.

Therefore,theset

U,,__ ^s,,

^isdensein

[c’, d’],

^acontradiction.

Theconsiderationof subcases 1, 2 and 3isconcluded andwereturn now tothemainproof.

Since

X

ishereditarily locallyconnected,it isthecontinuousimage ofanarcby

[12].

THEOREM 2. If

X

isasinTheorem1,then

(a) X

isrim-finite,

(b)

everysubcontinuum

G

of

X

has the property thatsomepointorapair of points separates

G,

and

(c)

eachclosedset irreducible with respect tothe property ofbeingacompactsetwhich sepa- rates

X

is metrizable.

PROOF.Theclaims

(a), (b)

^and

(c)

^followfrom

[19], [18]

^and

[4],

respectively, because Xcontainsnonon-degeneratemetric continuum.

Givenalocally connectedcontinuum

X,

for each pairofdistinct points a, bof

X

let

IX,

^a,

b]

denote the class of all continuous maps

f ^X

^{P such that P}

^f(X)

^{is anon-metricarc}

withend-pointsc anddand

f(a)

^cand

f(b)

^d. Also,introducearelation on

X

in the followingway: a b ifand onlyifa bor

[X,

^a,

b] .

THEOREM ^3.

Suppose

that

X

^is alocally connected^continuum. Then isanequiv- alence relationon

X,

and if

X

also satisfiesthe first axiom ofcountability, then equivalence classesof areclosed and theset

’

of equivalenceclassesof is upper semi-continuous.

PROOF. iseasilyseentobe reflexive and symmetric,sosupposethata b and b c

hold,but thatthereexists

f

^E

IX,

^a,

c]

^suchthat

f(X)

^{is anon-metricarc}

[d, e]

^with

f(a)

^d

and

f(c)

e.

CASE 1.

f(b)

^{d. Then}

f

^E

IX,

b,

c],

^acontradiction.

CASE 2.

f(b)

e analogous^toCase1.

CSE 3. d

<

y(b)

< . ^T

^{o oft5}

^[d,y(b)]

^d

^[y(b),]

^is non-mettle, so suppose

[d, f(b)]

^isnon-mettle. Define r

"[d,e] [d, y(b)]

othat

()

if xE

[d, f(b)]

^and

r(z) f(b)

^ifz

If(b), el.

^{Clearly, ro}

f . ^{[X,a, hi,}

^acontradiction.

Letusnowshow that each equivalence classG

andsuppose thatx E

-

^{G. ThereexistsacountablebasisU1,}

^U,...

^ofopenneighborhoods of x in

X

and a sequencex,z2,.., of points of G such that x, U, for 1,2,... Let

f" ^X

^--,

^[c,d]

^beacontinuous map ontoanon-metricarc

[c,d],

^where

f(zl)

cand

f(z)

^d.

Since each

[y(),y(z)]

is a metric subarc of

[c,d],

^it follows that

[c,d]

^is the closure ofa countable unionofmettleares. Consequently,

[c,d]

^{is separable, and} therefore mettlzable, a contradiction. ThusGisclosedinX.

Itremainstoshow that is upper semi-continuous if

X

^isfirstcountable. Lettheelement

G

of

’

^{beasubsetofanopen set}

^{U. Suppose}

^thatfor each open set

V

suchthat

G

C

V

C

U,

there isanelement

Gv

^{of sothat VIq}

Gv ^l

^and

Gv . ^U.

^{Thus, forsome pointx of}

^G

thereisa countablebasis

U, U,...

ofopenneighborhoodsof zsuch that for each

U,

^thereis

anelement

G,

of

"

^withtheproperty that

G,

^fq

U,

^}

Gi

^tq

(X U).

There is a pointy of

X U

sothat everyneighborhood of y intersects

Gi

for infinitely many i. Wemay assume withoutloss of generality that thereexists y, G,^[q

(X U)

^{for each} i, and that the pointsyi converge^{to y.} Letz, E

Ui

^{f’lG,for 1,2,...}

Thereexists

f [X,z, y]

^{such that}

f X [c,d],

^where

[c, d]

^isanon-metric arc,

f(x)

^c andf(y) d. Since the points

f(z,)

convergeto c,andthepointsf(yi)convergetod,andeach arc

If(z,), f(y,)]

ismetric,wefindthat

[c,d]

is metric acontradiction.

THEOREM 4. SupposethatXE

’’M

^[..J

’S

^[,..J

TMN

^andXisfirstcountable. Let be the family of allcomponentsofsetsin’. Then

X/TI

^isthecontinuousimage ofan arc.

PROOF. Since

c

is uppersemi-continuous, 7"/isupper semi-continuousas well

(see

e.g.

[28]).

Thus,7isaupper semi-continuous decomposition of

X

intoclosed sets and the quotient space

X/TI

^isalocallyconnected continuum.

If

X/TI

^is hereditarilylocally connected, we apply the main result of

[12]

^{to obtain the}

desired conclusion.

Otherwise,in

X/7"t

^{there is}asubcontinuumCsuch thatCfails to beconnectedim kleinen at apoint P. Thereis thus anopen set Win

X/7"I

^{such that}

P

E Wbut the component of WNCcontainingPcontainsnorelatively open subset ofC containingP. Let

Q

denote the element of

c

containing P. Thereis aclosed subset Sof

X

such that S C

[J

^W-

Q

and S separates

P

from

bd([J ^W)

^{inX. Let :X}

X/7"I

^{denote the}natural mapand let

B (S).

Let

U

denote the component of

X/TI- B

which containsP. Using the factsthat is upper semi-continuousand that

Q

NS

},

weletR1, R2,... G1, G2,...

V, V2

be subsets of

X/7"I

,similarly^asinthe proof of Theorem 1, except for theadditional conditionthatnoelement of intersects

cl([,J R)

^and

bd([,J U). ’

Now, lets,s2,.., and t,t2,.., be such that s, E

([.J

^G,)g

(Ubd(R1))

^andt, E

(UG,) ( bd(U))

^for 1, 2,... Since

X

isfirstcountable, wemayassumewithoutlossof generality that thepoints s, convergetosomepoints,andthe pointst, convergetosomepointt,and the limitingset

L

of

[.J G, U ^G,

^{G3, isacontinuumcontainingsandt.}

Thereisan

f ^{e [X,s,t]}

^{such that}

f(X)is

^anon-metric arc

[c,d]

^with

f(s)

^cand

f(t)

^d.

Wemaynowobtainacontradictionasinthe proof of Theorem 1.

ACKNOWLEDGMENT. H.M.Tuncali waspartially supported byan

NSREC

grant.

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