HOUSTON JOURNAL OF MATHEMATICS, Volume 6, No. 3, i980.

SOME LIMIT PROPERTIES OF C-MOVABLY REGULAR CONVERGENCE

Laurence Boxer

HOUSTONJOURNAL

OFMATHEMATICS

HOUSTON JOURNAL OF MATHEMATICS, Volume 6, No' 3, i980

SOMELMITPROPERTIESoFC-MOVABLYREGULARCONVERGENCE

Laurence Boxer

ABST-RACT' t€t dfio be the metric of C-movably regular

convergence lor the hyperspace mofX) of nonempty C-movable

subcompacta of a metric space X, recentiy introduced by Cerin'

We provide several answers, depending on the choice of property

o, the class C of topological spaces, and lhe nature of Ao' to the

question: Il .1irn dfio(An.Ao) = 0 and An e a for n = 1'2'3'"'' is

it true that"{€il in particular, we are concerned with

properties a loi which the analogous question' obtained by

replacing d$,, with Borsuk's fundamental metric dp' has an

affirmative answer.

l. h.rtroduction. Let 2X denote the collection of nonempty compact subsets of

ametricspace(X,d).lnrecentpapers,Borsuk[3]andCerin[8]havedefinedmetrics

ibr (subsets o0 2X that induce stronger topologies than that induced by the

rvell-known Hausdorff metric dg. several sirnilarities and dillerences between Borsuk's

'tiuldnmetltal metric (lF and Cerin,s metlic of C.movabb, regLllar Convergence dgo

were shown in t8l. Several more similarities and differences are shown in tl-tis paper'

There are several pfoperties a for which answers to the following question are

known (t3l , t5l, t9l ):

\1.1) If Lim dF(An,A )= 0 and An€a for n= I'?',3,"'' is it true that Aoec?ftr* ^

For some of these properties, [8] answers the analogous question:

(1 .2) If ,]5iOfiotnn,Ao) = 0 anrl An e a for n = 1,2,3'. '' is it true tlnt Ao € a7

Tlreanswerto(1.1)wasshownin[5]tobenegativeifaisallowedtobean

arbitrary hereditary shape property. As pointed ont in [9], positive answels to (1 1)

can be obtained either by restricting Ao or by taking a to be a particular property'

Similarly.(1.2)doesnotingeneralhaveapositiveanswer,butpositiveanswefsale

possible when suitable restrictions are placed on Ao, a, and the class of topological

spaces C.

2.Preliminaries.ByANR(/4)wemeananabsoluteneighborhoodretractfortlre

313

3t4 LAURENCEBOXER

class of metrizable spacos. By ANR we indicate a compact ANR(rt). An AR is an

absolute retract for the class ofmetrizable spaces, compact or not-

A continuous function f is an e-map if f has domain and range in a metric space

(X,d) with sup{d(x,f(x))ix € X} ( e .

For A,B e 2X,XCM where M€AR, dF(A,B)=e if e is the infimum of those

nonnegative numbers for which there are fundamental seeuenr-es/= ifk,A,Bly,y and

g= {gt,B,A}y,y and neighborhoods U and V of A and B. respecrively, in M, such

that for almost all k, ftriU and gllV are e-maps.

It is shown in [3] that the choice of M is not importanr: if h: ur-agAn + N € AR

is an embedding, then ,l$on(a",,to) = 0 if and onlv if

,rl$dprtrr\)Jr(Ao)) = 0.

Let Cbea class of topoiogical spaces. Suppose M €.A.){Rr-Dfr and B C U C V C M

where U and V are neighborhoods of B in M. We s-rife CIU.\':B; if for every map

f: K+V of a member K of C into V, if W is a neighborhood of B in M then f is

homotopic in Uto a map gi K+W. BV t9l. B€lX isC-mowble if tbrsome(hence

for every) embedding of B into an ANR(IO lI. for each neighborhood U of B in M

there is a neighborhood V of B inM with VCU sur-h rhar CIL-\-3i- If C= {A} for

some compactum A, this coincides with the notion of A<nor.abilir] t I I -

The subset of 2X consisting of its C-movable rnembers is denoted mofX). The

topology induced on mofX) UV Or9o is characterized in [E] bf-:

(2.1) THEOREM. Let XcM€ANR(.lf) and ter i\iFO c modX). Then

f3o,9.ro",Ao) = o if and ontv if(a) ligdg(An,Ao) = 0, and

(b) for every neighborhood U of A in M there is a neighbtrhood Y of A in M

such that Y C U and for almost a// n, C(U.V:A-1.

As with the topology of dp, the choice of M does nor affecr the topology of.Cumo'

3. Shape domination. Following [2], we say (Lr.\i.t is a neighborhood of (A,B)

in (M,N) if U is a neighborhood of A in M and V is a neighborhood of B in N. If M and

N are AR-spaces, A € 2M, B € 2N, and (U,V) is a neighborhood of rA,B) in (M,N),

then A and B are (U,Y)<quivalenl in (M,N) if there are fundamenral sequences/=

{fk,A,B}y,p and C = {ek,B,A}N,M and a neighborhood (Uo.Vo.} of {A,B) in (M,N)

SOME UMIT PROPERTTES OF CJO\/ABLY

such that for almost ali k' g1" fltUo- tUo *

the former homotopy holds for almost all k' ue

These notions are weaker than those of shape

resPectivelY.

Our first result strenglhens [5' 3'11 and has

(3.1) THEOREM' Let X be a merric 4

suppose {an}i=g C2x and f}*odAn-\o} =

Ao ln M, Lnand Loare (rJ '1J)-equivalefi

for rln

PR"ooF. Sinoe A= upoAn is compa'-t' t

Q is the Hilbert cube' Furthor' coover$eflt

convergence of {F(An)}pt to F(Ao) in the t-<

it follows that there is no loss of generality n sr

Let U be a neighborhood of Ao il

neighborhood of Ao in Q' There exists e ) 0 su

(3.2) lf Y is a topological space and h'li

h=k'Let N be a positive integer such that n )

Now we fix nZN' There exist fundamental

{e1,'to,'ln}Q,Q'a neighborhood (V'W) of t1

such that

(3.3) k 2 m implies ftrlV and 91lW are e

Also there is a neighborhood T of A" in Q ui

(3.4) V uW c P andkZm imPlies f1('

It follows from (3'3) and (3'4) that

(3.5) if k 2m then W U ft' 91(\!1 c P

T u ck'fs(T) c P and d(v'gs'f1(r)i <

By (3.2) and the fact that P C U' it follo*'s I

(3.6) k 2 m implies 1W = fk" Ell\[ in

It follows from (3.6) that A1, and Ao are {I-

An analogue holds for convergence in l

(3'7) THEOREM' SuPPose X C \[ €

SOME LIMIT PROPERTIES OF C-MOVABLY REGULAR CONVERGENCE 315

,R is an such that for almost ail k, gtr"fgiU6- 1Uo in U and ftr'9glVo- 1Vo in V' If only

theformerhomotopyholdsforalmostalti,wesayAis|J4ominatedbyBlnM,N.icspaceThesenotionsareweakerthanthoseofshapeequivalenceandshapedomination,

respectivelY.

ourfirstresultstfengthens[5,3.1]andhasasimilarproof.

(3.1) THEOREM' Let X be a metric space contained in the AR-space M'

-' r€ - tX and tim dpt An'Ao) =O' Then for each neighborhood U ofSuppose {Ani n=9 L / n€ -

-\o ln M, Anand Ao are (U'U)lequivalent for almost all n'

pROOF. Sinee A = uFOAn is compact' there is an embedding F: A + Q' where

Q ls the Hiibert cube' Further' convergence of {An}?=1 to Ao implies the

Jonvelgenceof{F(An)}p1toF(Ao)inthetopologyofdF.'Fromthisand[2,5.12j

1t follows that there is no loss of generality in simply assuming that M = Q'

Let U be a neighborhood of Ao in Q' Let P C U be a compact ANR

neighborhood of Ao in Q' There exists e ) 0 such that

G.2) lfY is a topological space and h'k: Y + P are maps that are e-close' then

h: k.

Let N be a positive integer such that n 2 N implies dp(A"'Ao) 1e 12 a.'d An c P'

\owweftxn2N.Thereexistfundamentalsequences/={fk,An,Ao}q,qandr=

ieg,.to,enlq,q, a neighborhood (V'W) of (An'Ao) in (Q'Q)' and a positive integer m

such that

(3.3) k 2 m implies fglV and 91lW are el2-maps'

o'so there is a neighborhood r of A. in e with T C v, and we may assume

(3.4) V uw c P andk2m implies fk(V) cP' gt(w) C V'and f1(T) cW'

It follows from (3'3) and (3'4) that

(3.5)ifk2mthenWUft"gl(W)cPandd(x'f1"e1(x))(eforallx€W;also

, u Bk fp(T) c P and d(v'gtr' fk(v)) <e for allv €T'

By (3.2) and the fact that P C U' it follows from (3'5) that

(3.6) k 2 m impiies 1W = fk'81lW in U and 11 : gO" fllT in U'

It follows from (3'6) that An and Ao are (U'U)-equivalent'

An analogue holds for convergence in the topology or ofrf;R'

(3.7) THEOR EM. suppose x c M € ANR(M), { A',}fr=o c mo4yg(X)' and

,f those

l,M and

,4, such

{EAR,0.

VCMy map

en f is

(hence

]inMA] for

). The

Then

inM

SY of

IA,B)

[. and

'{,N),)s /=M,N)

E

316 LAURENCEBOXER

(U,ll)-equivalent for almost all n.

PROOF. As in the proof of (3.1), we may assume ft = e.For a fixed neighborhood u of Ao in e, let ve -{\R be aneigJrborhood of Ao

in Q with v C u such that ANR(u,v;r{r. There is a posirire iateger m sucrr thatn ) m implies An C V and ANR(U,V;A'').

Let n2 m be fixed. There are sequences _-\-1;pg and {WI}F=O ofneighborhoods of Ao and Ar' respectively, in esuch that L- \-o =\t,o, V = Vt = Wt,k ) 1 implies Vtr and Wp are ANR's, niiOvt = eo" ,rf=gtrk = \, and for ail k,Vk*l c Vk, Wk*1 c Wk, ANR(Vk,Vk+1*4.j, and A\-Rrtt1-I[1*t l',).

Let ft =e1 = lq: Q * Q. We proceed inductir.ely. Sr:pp,ose l < k <i implieswe have maps fp,Bp: Q + e satisfying

(3.8) fk(D C Vp and s1(V) c W1.

Our choices of Vp and Wtr and (3.g) imply rhere are ma;rs F: \. X I + V1_1 andG: V X I - Wi_l (I denotes the unit inten-al.1 such that Fo = ri,\-- Fl(V) C Vi+l,Go = qlV, and G1(V) C Wi+l. We obtain maps \a1e;.l: e-e satistying (3.g) fork = i+l by taking fill and Bill to be extensions of F1 antl Gi - r*pectively.

We claim 1= {fk,An,Ao}e,q and s= .-e1_{o_\le-e *. fundamental

sequences. Let P be a neighborhood of Ao in e. There is a po'irir-e inreger i such thatVi CP. For k)i, fklv=fk+1lV in Vp_1 C Viap..o.f h a nrndamental sequence.

The proof that g is a fundamental sequence is similar.

It is easiiy seen that ror arl k, the following homoropies tuke piace in u:91" fLlV = e1 " lLlV = fplV - f1 lV = 1y and f1. 81,\- - fl, Bt \- = gk,\' = Cl lV = 1V.

This concludes the proof.

The following is an analogue of [5, 3.1 1 ] and a rheorem oi [9] :

(3.9) THEOREM. Let X be a metric space and let ..\rFO be a sequence inmo4yB(X) such that

"lLoANofonAJ= o. rf Ao is catm l7l. then sh(An)>

Sh(Aoi for almnst all n.

PROOF. Since the class P of all finite polyhedra is u-onrained in the class ANR, itfollows from [6, 4.2] that every member of mo41gq(X,] is a movable compactum.

Since Ao is also calm, we have Ao C FANR [9] .

SOME LIMIT PROPERTIES OF C-\{OVABLY R

Observe that (3'7) implies that ii L is a n'ei

t-dominated by An in Q'Q for almost all n' It ro

Sbr \) > Sh(Ao) for almost all n'

An immediate consequence of (3'9) is:

(3.i0) COROLLARY' If C= ANR arrir' '\o

rrstter for every hereclitary shape propert)"

The following theorem of [8] shorvs tlral t

ihei of dSo:

(3.11) THEOREM' ter {A,1iFOc mod

rm df"r An.Ao I = 0':rFai".rtu

a movable compactum is C-morablt

.-i.l l) that tlie example oi t5l' in rvhicll tlnit:

rl.o provides an example to show the results

.as>umption that Ao is calm is dropped' That a C:

:::t that a calm compactum has finitely man)' 'oI

4. Passage of a to limits' ln this se'-tion' v

:hoices of (Ca) without making additional a:st

rierv of (3.11), (i'2) is of interest primanll. tc

positive answer' Although (1'1) was originalll' st:

I ls a hereditary shape property' several resull

;roperties that are not hereditary shape properti'

For the property a of being quasi-dominai

tositive answer to (1'1) is [3,8'i] ' Similarll:

(4.1) THEOREM' SuPPose {A't}Po c o

1/ An is quasi-t)ominated by a cottlpd'i'!l

qtusi-dom)nated bY B'

PROOF. Let U be a neighborl-rood oi Ao

be an AR-space containing B' Frorn (3'lt ii lo

Udominated bv An in M'M' Bv []' 7'11 theri

rhat if C is a compactum in an AR-space T sit'

then Ao is U-dominated by C in N{'T- Tlie qu'

\o are

of Ao

h that

=0 0f

= w1,

all k,

mplies

.1 and

vi+1,

8) for

nental

:h that

uence.

in U:

= 1v.

nce tn

An) )

NR, it

tctum.

SOME LIMIT PROPERTIES OF C-IVIOVABLY REGULAR CONVERGENCE 3I'7

Observe that (3'7) implies that il U is a neighborhood of Ao in Q then Ao is

L-lominatedbyAninQ'Qforalmostailn'ltfollowsfrom[5'3''land3'10lthat

5r-, \) > Sh(Ao) for almost all n'

An immediate consequence of (3'9) is:

(3.10) COROLLARY' 1/ C=ANR and Lo is calm' tlten (1'2) has a positive

*'-';J',:1:-::::::':"::':::":"1' tr'ut rhe topologv or dp is stronger than

t"' ,ll,fi'rHEoREM. rer { An}Fo c rno6(X) ,, lgdr{An,Ao)

= 0' then

'r dfio(An,Ao)=0' --+.._" ic r_movable for every class c. lt follows from

]- Cl.u,1y a movable compactum is C-mov

-:.i1) that the example of t5l ' in which finite sets converge to a Cantor set in dp'

r.soprovidesanexampletoshowtheresultsoi(3.9)and(3.10)failwhenthe

usumption that Ao is caim is dropped' That a Cantor set is not calm follows frorn the

::;t that a calm compactum has finitely many components [7' 4'61 '

4.Passageofatolimits'lnthissection'weanswer(1'2)lorseveralparticular

::-oicesof(C,e)withoutmakingadditionalassumptionsaboutthenatureofA.,.In

.::rvof(3'11)'(1'2)isofinterestprimarilyforpropertiesaforwhich(1'1)hasa

psitiveanswel.Although(1.1)wasoriginatlystatedin[3]undertheasstrmptiontlrat

eisahereditaryshapeplopefty,several,.s.,tt,or[9]answer(i.1)positivelylor

:ioperties that are not hereditary shape properties'

For the property e of being quasi-dominated by a fixed compactum B i2l ' the

*"":; llTil:il': jl;l,l',i* []o' *"o* *' x) and;+-'*f t1"""] =

1 "

1l An is quasi-dominate(l by a compactum B for n-'1'2'3'"'' then Ao ls

'*":i:#"i?: ""a

neighborhood oi Ao ,' * :i ::::: T ; I ffiil,t^::'ne an AR-space containing B' From (3'?) it foilows that there ts

U4ominated by An in M'M' By l2''1'llthere is a neighborhood V of A'tinMsuch

ihatifCisacompactuminanAR.spaceTsuchthatAnisV.dominatedbyCinM,T'

rhen An is U-dominated by C in M'T' The quasi-domination of A" by B implies An is

318 LAURENCEBOXER

V-dominated by B in M,P. Our choice of V implies Ao is Udominated by B in M,p.

Since u is an arbitrary neighborhood of Ao in M, it follows rhat Ao is quasidominated

bv B.

Recall that a compactum L is C-ti-,-ial t9l jf for some (hence for every)

embedding of A into M € ANR(M), for each neighborhood U of A in M there is a

smaller neighborhood v of A in M such that ever-r- map of a member of c into v is

null-homotopic in U. We will use the following:

(4.2) THEOREM. t8l . Let {Arr}FO C tX -rnd /er Ao be C-trivial. Then

Ao e 1o6{X)

. and if ,lgdUtAn,a o) = O, then \ = mo6rxt for almost all n and

,lgo,!rotan'Ao) = o'

If C and D are classes of compacta, C slwpe dominttes D if for every X CD there

is a Y € Csuch that Sh(Y) > Sh(X).

In the following, S will denote a one-point space. a-rrC FAR will denote the class

of all compacta that are fundamentai absolute retrilc-ts-

(4.3) LEMM A. Every compactum ls FAR-rrir"i.zl .nd F-\R-mo,-able.

PROOF. It is obvious that every compacrun is iS;+riviat. Since {S} shape

dominates FAR, [6, 3.4] implies eyery conpiwtirn is F-{R-triyial.

It follows from the first conclusion of r 4.1 I that every compactum is

FAR-movable.

(4.4) COROLLARY. The identitl. map i: r lx-dn) * mop4p(X) rs a

homeomorphism

PROOF. By (4.3), i is a bijection. fet i\--fu c 2X. We must show

,lgouf,tn,ao) = 0 if and oniy tf Jtgt dI*R(AnAo) = 0. But this foliows from (2.1),

(4.2), and (4.3).

(4.5) COROLLARY. Let aobe any property mtisf1ing:

(.a) if A is a finite compactum then A € ao ord

(b) there is a compactum B such thnt B $ o'o.

Then for a = ao and C = FAR, ( I . 2 ) has a negat i''-e ailswer"

PROOF. This follows from (4.4), since B is the limit - dRtR of a sequence of

its finite subsets.

For example, (i.1) has a positive answer if a is movability [3,9.1] or nearly

SOME LIMIT PROPERTIES OF C-\tO\.ABLY I

tr-mrability [9]. Since soienoids are neither mc

mhr::g B to be a solenoid in (4'5) yields negative a

T-nere is a positive answer to (1'1) for the proPerr].

mh;ng B to be any positive-dimensional compar-tu

1r r i-l) for a = zero-dimensionality'

A closed subset B of a metric space X Li an

,rnil etery neighborhood U of B in X there is a ma1

Cr*erl1 any retract of X is an a-retract of X' \\e ha

r -1.6) THEOR EM' Suppose B € lX antl B €

B :; ; retract of X.

PROOF. Since B is a compact ANR' there e

1' .rf B in X onto B and [4, V3'i' page 10-1] 3n

i--i-: C+B are e-close, and f1 extends to a m

F.-\+B.There is a neighborhood U of B in X such t

srtisft' 0 < 6 < e12 andN6(B) c U'

Since B is an a-retract of X' there is a map

:hoi.-e of 6, f(B) CU' It follows that ibr I

.ruirxr.r) I el}+ 6 (e' By taking f1 =r'f B- i

:e;rrct of X.

ll a is the property of being an a-retra't o

llj. On the other hand, (1'2)has a negati\.e an

rire lbllowing shows:

(4.7) EXAMPLE' There is a sequence :--\

euclidean plane) such lhar irgd#(\-{or

\ rs ait a-retract of EZ for n = 1 '2 '3 " " '

PROOF. Let S1 be the unit circle rn El

arcs in Sl whose limit in dg (and theret-ore- b'

not an a-retra ct of E2 ' However' each of -{ t -{:

We remark that (4'5) and (4 7') iliu'trai: I

fail if the class used does not contain 'l\R' *r'

SOMELIMITPROPERTIESoFC.M0VABLYREGULARCONVERGENCE3I9

;-rnorability [9]. Since solenoids are neither movable nor nearly 1-movable [10],

rrnking B to be a solenoid in (4.5) yields negative answers lor these properties to (1.2).

llbere is a positive answel to (1.1) for the property of zero-dimensionality [9] ' but by

:&Ling B to be any positive-dimensional compactum in (4.:,) we get a negative answer

t:,1.1) for a = zero-dimensionality.

A closed subset B of a metric space x is an a-retracr of X [9] if for every e ) 0

moj e\-ery neighborhood U of B in X there is a map r: X + U such that rlB is an e-map.

. Then C-early any retract of X is an a-retract of X- We have:

,nand (4.6) THEOREM. Suppose B€2X andB€ANR7J Bisana-retractofXthen

D there

W:s r retract of X.

PROOF. Since B is a compact ANR, there exist a retraction r of a neighborhood

'r' oi B in X onto B and [4, v3.i, page 103] an e )0 such that if c is closed in X,

:--i.: C+B are e-close, and f1 extends to a map F1:X-+ B, then f2 extends to

F': -\+B.There is a neighborhood U of B in X such that U C V and rlu is an el2-map.Let

f rrtisfy 0 <6 < e12 andN6(B) c U.

since B is an a-retract of X, there is a map f: X -+ u such that flB is a 6-map. By

itroir-e of 6, f(B) CU. It follows that for x€B, d(r'l(x),x){ d(r" f(x)'f(x)) +

*itr).x) 1 el2+6 (e. By taking fl = r. flB, fz= ls,our choice of e implies B is a

reirac-t of X.

te class

in M,P.

minated

: every)

tere is a

rto V is

shape

um is

lia

show

(2.1),

ce of

early

If a is the property of being an a-retract of an AR X, (i.1) has a positive answef

-yl- on the other hand, (1.2) has a negative answer for this ploperty if c= FAR, as

:re foilowing shows:

(4.7) EXAMpLE. There is asequence {An}p0 c mop4B(E2)(whereE2 is the

euclidean plane) such rftar lgdffi(An,Ao)=0,Ao is not an a-retract of E), attd

Aois an a-retract of E2 for n = I ,2,3,-.. '

PROOF. Let 51 be the unit circle in E2 and let ,A'1,A2,A3,"' be a sequence of

.irr-SinS1 whoselimitindg(andtherefore,by(4.5),indI*R)isSl.Bv(4.6),S1 is

rot an a-retra ct of E2 .However, each of A 1 ,A2 'A3," '

is an a-retract of E2 '

Weremarkthat(4.5)and(4.?)illustratehowtheconclusionsof(3.9)and(3.i0)

iril if the class used does not contain ANR, even when Ao is still assumed to be cahn'

320LAURENCE BOXER

REFERENCES

,r r*' on a me ftization of the nv prrrpor" ; ; ;;'.;;,' |;ilr.' #j;., e4(1s7 7)

4. Theory of Retracrg polish Scientific publishers, Warsa w, 1967.5'

l;'ll1?lili,1fff B"^'k'' r';;;;;;;;";;"" ad stry domination,Bu, Ac polon

1.

2.

3.

6.

7.

8.

9.

10.

Muhlenberg CollegeAllentown, pennsylvania

1B I 04

\ " -,*

., f:; r,,:i: :: :: : :y: : : r o :

e :

t i e s,Ann. poron. Math.. 2s (1 e 7 4), s 3 .s 6._,Some quantitative nrnno,r;)o ^; "r-^^'^ ;-": v-u!- n'ur-, z>\ry 14),83-i

on o m o t, ;. -,, :."0: : T: t " r..f s hop e s, F und _ Math -, g3(t 97 6), p ; _; ; 2.

h::{#:3:ri{r{;':.r;:;;'^cattv compact swces at inrorite - triviatity and movabitity,

Homoropy ,"O"Jji? _?l locally compct st@ @t infinity _ catmnesssmoothness, pacific J. Math., 79(1978), 69_91 .

_ ^-_ , C-trnvabty regular convergence, preprint.

3 i:ffi;i,*, ;!,,)!; i :::, ;;; ;; ; ; ; ; ;., n,,*,,*,,"r me,ric, pr e p r int3; ;;YITH'1r','f,:;f r:::",'**,,'**i?#ffi HiiJiiiii'r.,Charlotte, i974, Studies in,

('v(c"'et 'us mree-ma"i]

r^-L^_- ^ opology, Academic pres, (1925),367_3g1.

and

Conf. at

Received July 30,1979