Homogeneous Plane Continua - University of Houston

HOUSTON JOURNAL OF MATHEMATICS, Volume 1, No. 1, 1975.

HOMOGENEOUS PLANE CONTINUA

CharlesL. Hagopian

ABSTRACT. A spaceis homogeneouisf for eachpairp, q of its
pointsthere existsa homeomorphismof the spaceonto itself
that takesp to q. R H Bing[3] provedthat everyhomogeneous
planecontinuumthat containsan arcis a simpleclosedcurve.F.
Burton Jones [9] showed that every homogeneous
decomposableplane continuumis eithera simpleclosedcurveor
a circle of homogeneousnonseparatingplane continua.Using
these results,we prove that every decomposablesubcontinuum
of a homogeneousindecomposablpelanecontinuumcontainsa
homogeneouisndecomposablceontinuum.It followsthat Bing's
theorem remains true if the word "arc" is replaced by
"hereditarilydecomposabcleontinuum."

1. Introduction. The simple closed curve, the pseudo-arc [2] [11], and the
circle of pseudo-arcs[4] are the only plane continuaknown to be homogeneousD. oes
there exist a fourth homogeneousplane continuum? A history of this unsolved
problem is given in [4] and [13]. According to Bing's theorem [3], if a fourth
homogeneousplane continuum exists,it does not contain an arc. Our generalization
(Theorem 2, Section 5) of Bing'sresult assertsthat sucha continuum cannot have a
hereditarily decomposablesubcontinuum.An example of a hereditarily decomposable
planecontinuumthat doesnot containan arcis givenin [ 1].

2. Definitions and preliminaries. In this paper, a continuum is a nondegenerate
compact connected metric space.A continuum is decomposableif it is the union of
two proper subcontinua; otherwise, it is indecomposable.A continuum is hereditarily
decomposableif all of its subcontinuaare decomposable.

A continuum T is called a triod if it contains a subcontinuum Z such that T - Z is

the union of three nonempty disjoint open sets.When a continuum doesnot contain a
triod, it is said to be atriodic.

A continuum is unicoherent provided that if it is the union of two subcontinuaE
and F, then E r3 F is connected. A continuum is hereditarily unicoherent if all of its

subcontinua are unicoherent.

35

36 CHARLES L. HAGOPIAN

LEMMA 1. If 34 is a homogeneousindecomposableplane continuum, then M is
atriodic and hereditarily unicoherent.

PROOF. Since M is indecomposable, it has uncountably many mutually
exclusive composants [12, Theorems 138 and 139, p. 59]. Suppose M contains a
triod. Since M is homogeneous,it follows that eachof its composantscontainsa triod.
This violates the fact that the plane does not contain uncountably many mutually
exclusivetriods [12, Theorem 84, p. 222]. Thus M is atriodic.

Suppose M contains a continuum F that is not unicoherent. Then each
composant of M contains a continuum that is homeomorphic to F. Note that F
separatesthe plane [12, Theorem 22, p. 175]. Since the plane is separableand M has
uncountably many composants,there exists a proper subcontinuumof M havingtwo
complementary domains that intersect M. This contradicts the fact that each
composant is dense in M [12, Theorem 135, p. 58]. Hence M is hereditarily

unicoherent.

3. Decomposition properties. Let D be a collection of compact subsetsof a
continuum M such that each point of M is contained in one and only one element of
D. The collection D is said to be a decompositionof M. If in addition, for eachopen
set Q of M, the set LJ{DCD' DcQ} is open, then Dis said to be an upper
semi-continuousdecompositionof M. Let k be the function that assignsto eachpoint
x of M the element of D that contains x. The function k is called the quotient map
associatedwith D. The collection D can be topologized by calling a subsetV of D

openif andonlyif k-1[V] isopenT. hespactehusobtaineisdcalletdhequotient

space associatedwith D.
Following E. S. Thomas [14], we define a continuum to be of type A provided it

is irreducible between two of its points and admits an upper semi-continuous
decomposition whose elements are connected sets and whose quotient space is the
unit interval [0,1 ]. Such a decomposition is called admissible.If a continuum M is of
type A and has an admissibledecomposition, each of whoseelements hasvoid interior
(relativeto M), thenM issaidto be of O'peA '.

LEMMA 2. Let M be a continuu•n that is irreducible between a pair of points. A

necessaraynd suj)qcientconditionthatM be of O•peA ' is that el,er)'subcontinuumof
M with nonvoidinterior be decomposabl[e10, Theorem•, p. 216] [14, Theorem10,

HOMOGENEOUSPLANE CONTINUA 37

p. 151.

LEMMA 3. If M is a continuum of type A, then M has a unique minimal

admissibledecomposition(relative to the partial ordering by refinement) [14,

Theorem3, p. $ and Theorem6, p. I 0/.

4. Homeomorphisms near the identity. A topological transfromation group

(G,M) is a topological group G together with a topological spaceM and a continuous

mapping (g,x)• gx of G X M into M such that ex = x for all x C M (½ denotes the

identity of G) and (gh)x = g(hx) for all g, h C G and x • M.

For eachx • M, let Gx betheisotropysubgrouopf x in G(thatis,thesetof all
g• G suchthatgx= x). LettingG/Gx betheleftcosestpacwe iththeusuatlopology,
themappinogfG/Gx ontoGxthatsendgsGx togxisone-to-onaendcontinuouTsh.e

set Gx is called the orbit of x.

Hereafter, M is a continuum with metric p and G is the topological group of

homeomorphismsof M onto itself with the topology of uniform convergence[10, p.

88]. E.G. Effros[5, Theorem2.1] provedthateachorbitisa setof thetypeG6 in M
if and only if for eachx•M, the mappingGx•g(x) of G/Gx ontoGx is a

homeomorphism.

LEMMA 4. Suppose M is a homogeneouscontinuum, e is a given positive

number, and x is a point of M. Then x belongsto an open subset W of M havingthe

following property. For eachpair y, z of points of W there existsa homeomorphismh

of M ontoM suchthath(y) = z and p(v,h(v)) • efor all v belongingtoM.

PROOF.SinceM is homogeneoutsh,eorbitof eachpointof M isM, a G6-set

HencethemappingTx: g-• g(x),beingthecompositioonf thenaturaol penmapping
of G ontoG/Gx anda homeomorphiosfmG/Gx ontoM, isanopenmappinogf G

onto M.

Let U be the open set consistingof all g G G suchthat p (v, g(v)) • 6/2 for each

v GM. DefineW to be the opensetTx[U]. Sincethe identitye belongtso U and
Tx(e)= x, thesetWcontainxs.

Assumye andz arepointsofW.Letf andgbeelementosf U suchthatTx(f) = y

andTx(g=)z.Sincfeix)=yandg(x=) z,thehomeomorph=higsfmflofMontoM

has the property that h(y) = z and ,o(v,h(v)) < • for all v G M. This completesthe

proof.

38 CHARLES L. HAGOPIAN

In [15], G. S.UngarusedthemappinTgx to provethatevery2-homogeneous
continuumislocallyconnectedF.orotherapplicationosfTx see[6] and[7].

5. Principal results.
THEOREM 1. SupposeM is a homogeneousindecomposableplane continuum
and A is a decomposable subcontinuum of M. Then A contains a homogeneous
indecomposable continuum.
PROOF. Since A is decomposable,there exist proper subcontinuaB and C of A
such that A = B •J C. Let b and c be points of B - C and C - B respectively.Let E be a

continuum in A that is irreducible between b and c.

The continuum E does not have an indecomposablesubcontinuum with nonvoid
interior (relative to E). To see this assumethe contrary. Let I be an indecomposable
continuum in E that contains a nonempty open subsetQ of E. SinceM is hereditarily
unicoherent and I is indecomposable, I is contained in E (3 B or E (3 C. Assume
without loss of generality that I is a subcontinuum of E (3 B. Let F be the
c-component of E-Q. Since F is a continuum that does not contain I and M is
hereditarily unicoherent, F meetsonly one composantof I. Let x be a boundary point
of Q that belongsto F. By Lemma 4, there exist homeomorphismsf and g of M onto
M (near the identity) suchthat (1) x, f(x), and g(x) belong to distinct composantsof I,
(2) { c, f(c), g(c) } is a subset of M - B, and (3) Q is not contained in f[Fl •J g[F].
Note that F, f[F], and g[F] are mutually disjoint. It follows that B •J F•J
f[F] U g[F] is a triod, which contradicts Lemma 1. Hence every subcontinuumof E
with nonvoid interior is decomposable.

Accordingto Lemma2, E is of typeA'. By Lemma3, E hasa uniqueminimal

admissible decomposition D, each of whose elements has void interior. Let k:
E -->[0,1 ] be the quotient map associatedwith D.

Thereexistsa numbesr(0<s < 1) suchthatk-l(s)isnotdegeneraftoer;

otherwise, E would contain an arc [ 14, Theorem 21, p. 29] and M would be a simple
closedcurve [3], which contradicts the assumptionthat M is indecomposable.Let Y

denottehecontinuukm-l(s).

Let p and q be distinct points of Y. We shall prove that Y is a homogeneous
subcontinuumof A by establishingthe existenceof a homeomorphismof Y onto itself
that takes p to q.

HOMOGENEOUS PLANE CONTINUA 39

Letr andt benumbesruscthhat0<r<s<t< 1.Defineetobep(k-l[[r,t],
k-l(0)U k-l(1)).

Let W be an open cover of Y such that for each W G W_•if y, z G W, then there

existsa homeomorphismh of M onto M suchthat h(y) = z and p (v, h(v)) < c for all

v• M (Lemma4). SinceY isa continuumth, ereexistsa finitesequenc{eWi} •=1of
elemenotsfWs_uchthatq• W1,p&Wn,andWi • Wi+1• • for1•<i • n.

Choo{sePi}•=0sucthhaPt 0=q,Pn=P,andPi&Wi(•Wi+lfor0• i • n.For

eachi (1 •<i •<n), lethi beahomeomorphisomfM ontoM suchthathi(Pi)= Pi-1and

p (v,hi(v))• c forallv GM.

Eachhi mapsY intoitself.Toseethis,assumhei[Y] isnotcontaineidnY. Note

thathik'l[[r,t]] doensotmeekt -l(0U) k'l(1).SincYe(•hi[Y]• ½andM is
atriodaicndhereditaurinlyicoherhein[Yt,]C hik-[1[r,t]]CE.Letd ande be

numberssuchthat khi[Y] = [d,e]. SinceE is irreduciblebetweenb andc, each

elemenHto=f(k-l(u)'d<u•e} isinhi[Y].
Becauhsei[Y](•hi[k-l(rU)k-l(t)]=3, thesethi[k-l)(ruk'l(t)] is in

k-l[[0,dU] [e1, ]].LetRandT bethecomponenhtis[Eo]f-hi[Yt]hact ontain
hik-l(r)andhik-l(t)respectivNeolyte.thatRUT isin k-l[[0,dU] [e1, ]]U
hik-l[[0],Ur [t,1]].Sincehik-sl(erp) arateshik-lf[r[o0m,hr)i][Y]inhi[E]a, nd
sincheik-l(ts)eparahtieks'[l(t,1]] fromhi[Y]inhi[E]t,heclosuorfeRUT does

notmeetanelemenot f H_H. encehi[Y] containasnonemptoypensubseotf hi[E],

whicchontradtihctesfacthathi[Y]isanelemeonf{t hik'l(uß)0•<u•<1} , the

minimaal dmissibdleecompositionf hi[E]. It followsthathi[Y] isa subseotf Y.

Usingthesameargumenot,necanshowthatif Y isnotcontaineidnhi[Y], then

Y has nonvoid interior relative to E, which contradicts the fact that Y is an element of

D. Thushi[Y] containYs. Consequentelya,chhi mapsY ontoitself.
It followsthathlh2 "- hnlY is a homeomorphisomf Y ontoY thattakesp to

q. Hence Y is homogeneous.

Since Y is a unicoherenthomogeneousplane continuum,it is indecomposable[9,

Theorem 2].

COROLLARY. If M is a homogeneousindecomposableplane continuum, then

no subcontinuum of M is hereditarily decomposable.

THEOREM 2. The simple closed curve is the only homogeneousplane

40 CHARLES L. HAGOPIAN

continuum that hasa hereditarily decomposablesubcontinuum.

PROOF. Let M be a homogeneousplane continuum that has a hereditarily

decomposablesubcontinuum H. According to the preceding corollary, M is

decomposable.

AssumeM isnot a simpleclosedcurve.Let G__bJeones'decompositionof M (to a

circle) having the property that each of its elements is a homogeneous,

indecomposablneo, nseparatinpglanecontinuum[8, Theorem2] [9, Theorem2]. It

follows from our corollary that H doesnot lie in one elementof G. However,if H

meets more than one element of G, then H containsan element of G [9, Theorem 1],

which contradictsthe assumptionthat H is hereditarily decomposableH. ence M is a

simple closedcurve.

ADDED IN PROOF. In a subsequenpt aper,the authorhasimprovedTheorem 1

by provingthatnosubcontinuumof ahomogeneouinsdecomposabplelanecontinuum

is decomposableIt. followsthat everyhomogeneounsonseparatinpglanecontinuumis

hereditarily indecomposable.

REFERENCES

1. R. D. Anderson and G. Choquet, A plane continuum no two of whosenon-degenerate
subcontinuaare homeomorphic: An application of inverselimits, Proc. Amer. Math. Soc.
10(1959),347-353.
R H Bing,A homogeneouinsdecomposabplelanecontinuumD, ukeMathJ. 15(1948),729-742.
, A simpleclosedcurveis the only homogeneousboundedplane continuumthat contains
anarc,Canad.J. Math. 12(1960),209-230.
R H Bingand F. B. JonesA, notherhomogeneoupslanecontinuum,Trans.Amer. Math. Soc.
90(1959), 171-192.
E.G. Effros,TransformatiognroupsandC*-algebrasA,nn.of Math.81(1965),38-55.
C. L. HagopianA, fixed-pointheoremfor homogeneocuosntinuaM, ichiganMath.J.21(1974),

233-234.

,The fixed-pointpropertyfor homeomorphismosf almostchainablehomogeneous
continua(,to appear).
F. B. JonesC, ertainhomogeneouusnicoherenitndecomposabcloentinuaP, roc.Amer.Math.
Soc.2(1951), 855-859.

, Ona certaintypeof homogeneopulasnecontinuumP,rocA. mer.Math.Soc.6(1955),

735-740.

10. K. KuratowskiT, opologyVol. 2, 3rd Ed., MonografieMat., Tom 21, PWN,Warsaw1, 961;
Englishtransl.A, cademiPcressN, ewYork;PWN,Warsaw1,968.

11. E.E. MoiseA, noteonthepseudo-arTcr,ansA. mer.Math.Soc.67(1949),57-58.
12. R. L. Moore,Foundationosœpoinst ettheory,Rev.Ed.,Amer.Math.Soc.Colloq.Publ.,Vol.

13, Amer.Math. Soc.,ProvidenceR, . I., 1962.

13. J. T. RogersJ,r.,Thepseudo-ciricslneothomogeneoTursa,nsA. mer.Math.Soc.148(1970),

HOMOGENEOUSPLANECONTINUA 41

417-428.

14. E. S.ThomasJ,r.,MonotondeecompositiofnirsreducibcloentinuaR,ozprawMyat.50(1966),

1-74.

15. G.S.UngarO, nallkindsofhomogeneosupsaceTsr,ansA. merM. ath.Soc(.toappear).

CaliforniaState University ReceiveAdpril4, 1975.

SacramenCtoa,liforni9a5819