Continuous curves of nonmetric pseudo-arcs and semi-conjugacies to interval maps

0.19 MB·

by  Jan P. Boronski

CONTINUOUS CURVES OF NONMETRIC PSEUDO-ARCS AND
SEMI-CONJUGACIES TO INTERVAL MAPS
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1
JANP.BORON´SKIANDMICHELSMITH
0
2

n Abstract. In1985M.Smithconstructedanonmetricpseudo-arc;i.e. aHaus-
a dorffhomogeneous,hereditaryequivalentandhereditaryindecomposablecon-
J tinuum. Taking advantage of a decomposition theorem of W. Lewis, he ob-
7 taineditasalonginverselimitofmetricpseudo-arcswithmonotonebonding
1 maps. Extending his approach, and the results of Lewis on lifting home-
omorphisms, we construct a nonmetric pseudo-circle, and new examples of
] homogeneous 1-dimensionalcontinua; e.g. acircleandsolenoidsofnonmetric
N pseudo-arcs. Among many corollaries we also obtain an analogue of another
theorem of Lewisfrom1984: any interval mapissemi-conjugate toahomeo-
G
morphismofthenonmetricpseudo-arc.
.
h
t
a
m
1. Introduction
[
The present paper is motivated by a well-known open problem in topology.

2
v Problem: Classify homogeneous compacta.
2
6 MisloveandRogersshowedin[23] thateachhomogeneouscompactmetric space is
8 theCartesianproductofahomogeneouscontinuumY andthesetC,whereC isthe
1
Cantor set, or a finite space. Recall that a compact space X is homogeneous if for
0
any two points x and y in X there exists a homeomorphism h :X →X such that
.
1 h(x) = y. Among the well-known homogeneous compact spaces there are objects
0
such as the circle, Cantor set, surfaces of the sphere and torus, and 1-dimensional
7
Vietoris solenoids. Finite spaces are homogenous in a trivial way. Despite the fact
1
: that some spaces are not homogeneous, they are very close to being so and in this
v
case it is useful to look at their degree of homogeneity. We say that the degree of
i
X homogeneity of the space X is 1/n, where n is a cardinal number, when the group
r of homeomorphisms of X has exactly n orbits; i.e. X has exactly n topologically
a
distinct types of points. A classical nontrivial example of a 1/2-homogeneouscon-
tinuumistheSierpin´skicarpet[19]. Thereisanothergroupofcompactspacesthat
areveryclosetobeinghomogeneousinthesensethattheirhomeomorphismgroups
act minimally on them. Namely, we say that X is nearly homogeneous when for
any pair of points x and y in X and any open neighborhood U of y there exists a
homeomorphism h : X → X such that h(x) belongs to U. There are examples of
spaces that are nearly homogeneous, although their degree of homogeneity is 1/n
for no n∈N. One such example is a compactum that belongs to a class of heredi-
tarily indecomposable continua, particularly important from the view point of this
study. We say that Y is an indecomposable continuum if it cannot be represented
2010 Mathematics Subject Classification. Primary54F15;Secondary 54F65.
Key words and phrases. homogeneous, pseudo-arc, pseudo-circle, continuous decomposition,
Whitneymap,curve,semi-conjugacy.
1
2 JANP.BORON´SKIANDMICHELSMITH
asthesumoftwopropersubcontinua. ContinuumY ishereditarily indecomposable
if every subcontinuum is indecomposable. Two particularly prominent examples
of hereditarily indecomposable continua are a pseudo-arc and pseudo-circle, some-
times referred to as bad fractals due to their very complex structure and certain
degree of self-similarity. Pseudo-arc is a space first constructed by B. Knaster in
the 20s of the XX century [18]. It is a one-dimensional planar continuum, that
does not separate the plane, and is homeomorphic to any of its subcontinua [24].
In the 40s R.H. Bing showed that the pseudo-arc is homogeneous [3] and later,
with Jones, he constructed a homogeneous (decomposable) circle of pseudo-arcs
[6]. Bing also constructed the pseudo-circle [4], the unique hereditarily indecom-
posable planar cofrontier. The pseudo-circle is nearly homogeneous [17], though
neither homogeneous [10], [27] nor 1/n-homogeneous for any n>1. Lewis showed
′
that for any 1-dimensional continuum Y there exists a continuum Y that admits
a continuous and monotone decomposition into pseudoarcs, and if Y is homoge-
′
neous then so is Y [22]. Most recently in [13] Hoehn and Oversteegen classified
all planar homogeneous continua as: the circle, pseudo-arc, and circle of pseudo-
arcs. It is noteworthy that the pseudo-arc appeared quite recently also in studies
concerning other areas of mathematics, such as functional analysis and logic. In
2005, Rambla [26] and Kawamura [16] showed independently, that the pseudo-arc
minus a point provides a counterexample to the Conjecture of Wood, formulated
intheearly80s,abouttheexistenceofcertainBanachspaceswithtransitivegroup
of isometries. About the same time Solecki and Irwin [15] obtained the pseudo-arc
as the quotient space of a projective Fr¨ıss´e limit of reflexive linear graphs. Ex-
panding their tools, in 2014 Kwiatkowska showed that the pseudo-arc has a dense
conjugacy class of homeomorphisms [20], reproving the result of Oppenheim from
2008 [25]. These findings further confirm the importance of researchon hereditary
indecomposablespacesandtheirsignificancebeyondanarrowareaofmathematics.
Moreover Smith’s nonmetric pseudo-arc has recently provided another counterex-
ample to Wood’s conjecture [8], which highlights the importance of including the
nonmetric hereditarily indecomposable continua in future studies. In the present
paper,buildingontheapproachofthesecondauthor,andheavilyrelyingonLewis’
results for the metric pseudo-arc, we extend Lewis’ results to prove the following.
Theorem A. For every one-dimensional metric continuum M, there exists a con-
tinuumM suchthat M has a continuous decomposition intononmetric pseudo-
ω1 ω1
arcs, and the decomposition space is homeomorphic to M. Additionally:
(1) If M is homogeneous, then so is M .
ω1
(2) Every homeomorphism of M lifts to a semi-conjugate homeomorphism of
M .
ω1
(3) M admits a Whitney map onto a long arc.
ω1
WealsogiveananalogueofanotherresultofLewis. In[21]heshowedthatanyself-
map of an arc-like continuum is semi-conjugate to a pseudo-arc homeomorphism.
In the present paper we exploit his result to connect one-dimensional dynamics to
the dynamics on Smith’s nonmetric pseudo-arc.
TheoremB.Everyintervalmapissemi-conjugatetoahomeomorphism ofSmith’s
nonmetric pseudo-arc.
CONTINUOUS CURVES OF NONMETRIC PSEUDO-ARCS 3
The paper is organized as follows. In Section 2 we recall some preliminary
facts, including the results of Lewis and properties of Smith’s nonmetric pseudo-
arc. In Section 3 we describe the construction of continuous curves of nonmetric
pseudo-arcs and discuss some curious applications of the method. One of them is
the aforementionedresult on the semi-conjugacy of interval maps with homeomor-
phisms of the nonmetric pseudo-arc. In Section 4 we show that our long inverse
limit approach cannot be continued beyond ω , as in a sense it stabilizes at this
1
step. In Section 5 we observe that any continuous curve of nonmetric pseudo-arcs
from Section 3 admits a generalized Whitney map onto the long arc. We conclude
our paper in Section 6 with some questions on Smith’s nonmetric pseudo-arc.
2. Preliminary results
RecallthatgivenagraphG,acontinuumX iscalledG-likeifforeachopencover
U ofX there is a finite openrefinement{U ,...,U } whosenerveis G. IfG =[0,1]
1 t
thenX iscalledarc-likeorchainable. BelowwerecalltheresultsofLewis,onwhich
we shall heavily rely.
Theorem 2.1 (Lewis [22]). Suppose that M is a one-dimensional metric contin-
uum. Then there exists a one-dimensional continuum Mˆ and a continuous decom-
position G of Mˆ into pseudo-arcs so that the decomposition space Mˆ/G is homeo-
morphic to M. Furthermore if π : Mˆ → Mˆ/G is the mapping so that π(x) is the
unique element of G containing x and h:Mˆ/G→Mˆ/G is a homeomorphism then
there exists a homeomorphism hˆ :Mˆ →Mˆ so that π◦hˆ =h◦π.
Thuswehaveanopenandclosedmapπsothatthefollowingdiagramcommutes:
Mˆ oo hˆ Mˆ
π π
(cid:15)(cid:15) (cid:15)(cid:15)
Mˆ/G=M oo h M =Mˆ/G
Theorem 2.2 (Lewis [22]). Under the hypothesis of Theorem 2.1, if for some
element g ∈ G we have x,y ∈ g then there exists a homeomorphism hˆ so that
hˆ(x)=y and π◦hˆ =π.
Theorem 2.3 (Lewis [22]). Suppose that X is a pseudo-arc and G is a continuous
collection of pseudo-arcs filling up X so that for each x ∈ X, π(x) is the element
of G containing x and Y = X/G. Then Y is a pseudo-arc, and if h : Y → Y is a
homeomorphism thenthereexistsahomeomorphism hˆ :X →X sothatπ◦hˆ =h◦π.
oo hˆ
X X
π π
(cid:15)(cid:15) (cid:15)(cid:15)
oo h
Y Y
Theorem 2.4 (Lewis [22]). Suppose X is a metric chainable continuum and f :
X →X is a map. Then there exists a pseudo-arc homeomorphism h:P →P and
4 JANP.BORON´SKIANDMICHELSMITH
a continuous surjection p:P →X such that f ◦p=p◦h. In particular, p gives a
semi-conjugacy between f and h.
The following result will also be essential to our proofs.
Theorem2.5. [9]SupposethatX =lim{X ,fβ} andY =lim{Y ,Gβ}
←− α α α<β<ω1 ←− α α α<β<ω1
are inverse limits and there is an index γ <ω and a collection of mappings ϕ for
1 δ
δ >γ the following diagram commutes:
X oo fγδ X
γ δ
ϕγ ϕδ
Y(cid:15)(cid:15) oo gγδ Y(cid:15)(cid:15)
γ δ
Then there is an induced mapping ϕ:X →Y so that the following commute:
X oo πγ X
γ
ϕγ ϕ
(cid:15)(cid:15) (cid:15)(cid:15)
Y oo πγ Y
γ
In [29] the second author constructed a nonmetric Hausdorff hereditarily inde-
composable arc-like continuum, as an ω -long inverse limit of metric pseudo-arc
1
with monotone, open and closed bonding maps, since then referred to as Smith’s
nonmetricpseudo-arc. Thefollowingtheoremlists someoftheknownpropertiesof
that space.
Theorem 2.6. Let P be Smith’s nonmetric pseudo-arc. Then
ω1
(1) P is homogeneous and hereditarily equivalent [29],
ω1
(2) P is separable [8],
ω1
(3) P is not first countable at any point [8],
ω1
(4) P contains a convergent sequence [8].
ω1
In the next section we are going to generalize Smith’s construction to other
metric continua.
3. Continuous curves of nonmetric pseudo-arcs
Theorem3.1. Foreveryone-dimensionalmetriccontinuumM,thereexistsacon-
tinuumM suchthat M has a continuous decomposition intononmetric pseudo-
ω1 ω1
arcs, and the decomposition space is homeomorphic to M.
Proof. Let M = M be a one-dimensional continuum. By Lewis’ decomposition
0
theorem there is a continuum M , a collection G of mutually disjoint pseudo-arcs
1 1
in M , and an open and closed map p : M → M such that x ∈ M if and only
1 0 1 0
if the point-inverse p−1(x) ∈ G ; furthermore if P : M /G → M is defined by
0 1 0 1 1 0
P (g)istheuniqueelementofM sothatp−1(x)=g thenP isahomeomorphism.
0 0 0 0
Suppose that n > 1 is an integer and Mn,Gn,pn−1,Pn−1, have been con-
structed. Then, by Lewis’ theorem, there is a continuum M , a collection
n+1
G of mutually disjoint pseudo-arcs in M , and an open and closed map
n+1 n+1
p :M →M suchthatx ∈M ifandonlyifthepoint-inversep−1(x )∈G
n n+1 n n n n n n+1
CONTINUOUS CURVES OF NONMETRIC PSEUDO-ARCS 5
and if P :M /G →M is defined by P (g) is the unique element of M so
n n+1 n+1 n n n
that p−1(x) = g then P is a homeomorphism. For each pair of integers m < n
n n
define pnm = pm ◦pm+1 ◦...◦pn−1; let Mω = ←lim−{Mn,pnm}m<n<ω and define pωn
th
to be the projection of M onto the n coordinate space. Observe that if m < n
ω
then pω =pn ◦pω and that as a projection the map pω is closed and open.
m m n n
We continue by transfinite induction to construct M for every α < ω as in
α 1
[29]. Suppose that α is an ordinal and that M ,G ,pβ, have been constructed for
β β γ
all γ <β <α.
Case 1. α is a limit ordinal. Then define:
M = lim{M ,pβ}
α ←− γ γ γ<β<α
and observe that
pα = pβ ◦pα
γ γ β
where pα :M →M is the projection map.
β α β
Case 2. α is not a limit ordinal. Since α is a countable ordinal, Mα−1 is a
one-dimensional metric continuum. Then by Lewis’ decomposition theorem there
is a continuum M , a collection G of mutually disjoint pseudo-arcs in M , and
α α α
anopenand closedmappαα−1 :Mα →Mα−1 suchthat x∈Mα−1 if and only if the
point-inverse pαα−1−1(x) ∈ Gα; furthermore if Pα−1 : Mα/Gα → Mα−1 is defined
by Pα−1(g) is the unique element of Gα so that pαα−1−1(x) = g then Pα−1 is a
homeomorphism. We define
pαγ = pαγ−1◦pαα−1.
Let M =lim{M ,pα} and let pω1 be the natural projection onto the αth
ω1 ←− α β β<α<ω1 α
coordinate space M . We now show that M is a continuum with the desired
α ω1
properties.
Claim 3.1.1. pω1 is open.
0
Proof. (of Claim 3.1.1) Since pω1 is a projection onto the coordinate space M of
0 0
the inverse limit, it is open. (cid:3)
Claim 3.1.2. pω1 is closed.
0
Proof. (of Claim 3.1.2) This follows from the fact that M is a closed subset of
ω1
the Cartesian product M and pω1 coincides with the projection onto the
γ<ω1 γ 0
0th coordinate, with M compact. (cid:3)
Qγ<ω1 γ
Claim 3.1.3. For evQery x ∈ M and 0 < α < ω the point-inverse pα(x) is a
0 1 0
pseudo-arc.
Proof. (of Claim 3.1.3) Fix x ∈ M . Then (p1)−1(x) is a pseudo-arc. Sup-
0 0
pose that α < ω and for β < α, pβ is a pseudo arc. For α a limit ordi-
1 0
nal (pα)−1(x) will be homeomorphic to lim{pβ−1(x),pα} . But this is
0 ←− 0 γ 0<γ<β<α
the countable inverse limit of hereditarily indecomposable chainable metric con-
tinua and so it must be a metric chainable hereditarily indecomposable continuum
6 JANP.BORON´SKIANDMICHELSMITH
as well, and so it is the pseudo-arc by its uniqueness. For α a successor ordi-
nal, (pα)−1(x) = (pα )−1 p(α−1) −1(x) is a continuous decomposition of the
0 (α−1) 0
pseudo-arc p(0α−1) −1(x)int(cid:0)o(cid:0)pseudo(cid:1)-arcsG(cid:1)α ={ pα(α−1) −1(t)|t∈ p(0α−1) −1(x)}
and so is a pseudo-arc by Theorem 2.3. (cid:3)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
ConsequentlyMω1 =lim←{Mα,pβα :α<β <ω1}is aspacewithanaturalopen
and closed projection map pω1 onto M , where for each x ∈ M the point-inverse
0 0 0
pω1 −1(x) is the nonmetric pseudo-arc constructed in [29]. For each x ∈ M
0 0
let gω1 = {p ∈ M |p = x}. Therefore if G = {gω1|x ∈ M } then G is a
(cid:0) (cid:1)x ω1 0 ω1 x 0 ω1
continuousdecompositionofM intononmetricpseudo-arcswhosedecomposition
ω1
space M /G is M . (cid:3)
ω1 ω1 0
It follows from Lewis’ construction that if M is G-like then his “M of metric
pseudo-arcs” is G-like as well. Because of the inverse limit construction it follows
that the nonmetric M will be G-like as well. Moreover, since pω1 has acyclic
ω1 0
fibers, by Vietoris–Begle mapping theorem [30], the 1st Cˇech cohomology groups
ofM and M will be isomorphic. An interesting consequenceis the existence of a
ω1
nonmetric pseudo-circle.
Corollary 3.2. There exists a nonmetric pseudo-circle C ; i.e. a hereditarily
ω1
indecomposable, nonchainable circle-like continuumsuch that its 1st Cˇech cohomol-
ogy group is Z. In addition, there is an open map πω1 : Cω1 → C, where C is the
pseudo-circle.
Theorem 3.3. If h:M →M is a homeomorphism then there exists a lift homeo-
morphism h¯ :M →M such that h◦π =π◦h¯, where π :M →M is the open
ω1 ω1 ω1
and closed quotient map.
Proof. We proceed by transfinite induction applying Lewis’ result for the metric
pseudo-arc to the above inverse limit construction. Let h : M → M be the given
homeomorphism. Then by Theorem 2.1 there exists a homeomorphism h :M →
2 2
M so that the following diagram commutes:
2
M oo h2 M
2 2
(cid:15)(cid:15)p21 (cid:15)(cid:15)p21
oo h
M M
Suppose that h has been constructed for all γ < α so that for β < γ we have
α
pγ ◦h =h pγ.
β γ β β
Case 1: α = γ +1 is a successor ordinal. Then by Theorem 2.1 there exists a
homeomorphism h :M →M so that the following diagram commutes:
α α α
M oo hα M
α α
pα pα
γ γ
(cid:15)(cid:15) (cid:15)(cid:15)
M oo hγ M
γ γ
CONTINUOUS CURVES OF NONMETRIC PSEUDO-ARCS 7
Since for β <γ we havepγ ◦h =h pγ; then this together with the abovegives us
β γ β β
pα◦h =h pα.
β α β β
Case 2: α is a limit ordinal. Then by construction M = lim{M ,pβ} .
α ←− γ γ γ<β<α
Defineh coordinate-wisebyh ({x } )={h (x )} . Thenbytheproperties
α α γ γ<α γ γ γ<α
ofinverselimits,h willbeahomeomorphismandforγ <αwehavepα◦h =h pα.
α γ α γ γ
Now we define h coordinate-wise by h ({x } ) = {h (x )} . Which
ω1 ω1 γ γ<ω1 γ γ γ<ω1
by the inductive formula that gives us pα◦h = h pα where h = h we will have
1 α 1 1 1
pω1 ◦h =h ◦pω1 as required. (cid:3)
1 ω1 ω1 1
Similarly, using Theorem 2.2, one proves the following. The proof is left to the
reader.
Theorem 3.4. If x,y ∈ M are such that pω1(x) = pω1(y) then there exists a
ω1 1 1
homeomorphism hˆ so that hˆ(x)=y and pω1 ◦hˆ =pω1.
1 1
Corollary 3.5. If M is homogeneous then so is M .
ω1
Corollary 3.6. There exists a homogeneous circle of nonmetric pseudo-arcs, and
homogeneous solenoids of nonmetric pseudo-arcs.
Corollary 3.7. The nonmetric pseudo-circle from Corollary 3.2 is nearly homo-
geneous.
The following corollary can be proved following the arguments presented here,
combined with those in the proof of Lemma 4.1 in [7].
Corollary 3.8. For every n > 0 there exists a 1-homogeneous solenoidal contin-
n
uum of nonmetric pseudo-arcs.
Proof. Fixn>0andletS beahereditarilydecomposable 1-homogeneoussolenoidal
n n
continuumconstructedin[14]. ThenbyLemma4.1in[7]thereexistsa 1-homogeneous
n
solenoidal continuum of pseudo-arcs. Now proceed to the long inverse limit. The
details are left to the reader as an exercise. (cid:3)
Theorem 3.9. Suppose X is a metric chainable continuum and f : X → X is a
map. Then there exists a homeomorphism of the nonmetric pseudo-arc h : P →
ω1
P and a continuous surjection p:P →X such that f ◦p=p◦h.
ω1 ω1
Proof. LetM bethepseudo-arcP,thenapplyingTheorem2.4thereexistsahomeo-
morphismh:M →M andacontinuoussurjectionq :M →X suchthatf◦q =q◦h.
Then from Theorem 3.3 there is a homeomorphism h : M → M such that
ω1 ω1 ω1
h ◦pω1 = pω1 ◦h . It is straightforward to verify that h will be the required
ω1 1 1 ω1 ω1
map. (cid:3)
4. More on decompositions of M
ω1
In this section we are interested if the procedure described in the previous sec-
tion could be extended beyond ω , as to produce other nonmetric pseudo-arcs and
1
correspondingnewdecompositions. Asweshallshow,thisisunfortunatelynotpos-
sible by the same approach, as any decomposition space of M must be a metric
ω1
continuum.
Theorem 4.1. Let M =lim{M ,fβ} bethe non-metric one-dimensional
ω1 ←− α α α<β<ω1
continuum constructed above. Suppose that G is a continuous collection of non-
degenerate continua that fills up M . Then M /G is a one-dimensional metric
ω1 ω1
continuum.
8 JANP.BORON´SKIANDMICHELSMITH
Proof. Recall: M is anarbitraryone dimensionalmetric continuum. Forα>0, α
0
a successor ordinal, Gα ={fα−−11(p)| p∈Xα−1} is the continuous decomposition of
Mα−1 used to construct Mα. We state and prove some claims concerning a single
element H ∈G.
Claim 4.1.1. Let H ∈Gand suppose thatα is the firstcoordinateso thatπ (H)
α
is non-degenerate. Then α is not a limit ordinal.
Proof. (of Claim 4.1.1) If α is a limit ordinal then M is the inverse limit of the
α
spaces {M } . Then if π (H) is non-degenerate, there must be a γ <α so that
γ γ<α α
fα(π (H)) is non-degenerate which contradicts the hypothesis. (cid:3)
γ α
Suppose that α is defined as in Claim 1; we are interested in the case where
α > 0. So suppose that πα(H) is a subset of fα−−11(p) for some point p ∈ Xα−1.
Then for γ < α, π (H) is, by definition, a singleton. For claims 2 and 3 suppose
γ
that H ∈G and α is defined as in claim 1.
Claim 4.1.2. π (H) lies in a single element of G .
α α
Proof. (of Claim 4.1.2) If πα(H) intersects two element of Gα then πα−1(H) is
non-degenerate;this contradicts the definition of α. (cid:3)
Suppose now that x = {x } ∈ H and consider an arbitrary γ. Then for
γ γ<ω1
γ < α we have x = π (H) since π (H) is degenerate in this case. For γ = α, γ
γ γ γ
is not a limit ordinal so xγ ∈ fγ−−11(xγ−1); though it is possible that there is some
point y ∈ M with yγ ∈ fγ−−11(xγ−1) so that y ∈/ H. Suppose that γ = α+ 1.
Then since α>0, by construction π (H) is a pseudo-arc and there are two points
α
x and y so that π (H) is irreducible from x to y . From the properties of
α α α α α
continuousdecompositionsofthepseudo-arc,wealsohavethefactthatf−1(π (H))
α α
is irreducible between each point of f−1(x ) and each point f−1(y ). So π (H)=
α α α α γ
π (H)=f−1(π (H)). Therefore, since π (H) is a union of elements of G
α+1 α α α+1 α+1
for γ >α+1 we have π (H)=fγ −1(π (H)). So this gives us:
γ α+1 α+1
Claim 4.1.3. H ={{x } | x ∈π (H)}.
γ γ<ω1 α α
Foreachnon-degeneratesubcontinuumH ofM letα denotethefirstordinal
ω1 H
αsuchthatπ (H)isnon-degenerate. Thenthefollowingfollowsfromthediscussion
α
preceding Claim 4.1.3:
Claim 4.1.4. If H ∈ G is a non-degenerate subcontinuum of M then for each
ω1
γ >α we have H =π−1(π (H)).
H γ γ
Claim 4.1.5. Suppose H ⊂M is a non-degeneratecontinuum. Thenthere is an
ω1
open set U in C(M ) containing H so that if K ∈U then α ≤α +2.
ω1 K H
Proof. (of Claim 4.1.5) Observe that the following collection is closed set in the
hyperspace C(M ) that misses H:
ω1
{K ∈C(M )|α ≥α +2}.
ω1 K H
(cid:3)
From Claim 4.1.5 and the compactness of M we have the following:
ω1
Claim 4.1.6. There exists an ordinal λ so that for each H ∈G we have α <λ.
H
CONTINUOUS CURVES OF NONMETRIC PSEUDO-ARCS 9
Consider now the collectionJ ={π (H)|H ∈G} where λ is the ordinalguaran-
λ
teed by Claim 4.1.6. Then from the above claims and the fact that all the relevant
maps are open, J is a continuous decomposition of M . This gives us the M /G
λ ω1
is homeomorphic to M /J. Since M is a one dimensional metric continuum and
λ λ
J isa continuousdecompositionofM into pseudo-arcs,itfollowsthatM /J,and
λ λ
hence M /G, is a one dimensional metric continuum. (cid:3)
ω1
5. Generalized Whitney maps on M
ω1
In [12] Hern´andez-Guti´errez proved that there is a generalized Whitney map
from the non-metric pseudo-arc of Smith onto the long arc. In the following we
generalize his result to the continua M constructed in Section 3.
ω1
SupposethatAisaHausdorffarcwithanorderrelation⊳generatingthetopol-
ogy; suppose further that a⊳b are the non-cut points of A. Suppose that X is a
continuum then the statement that µ : C(X) → A is a generalized Whitney map
means that µ is a continuous function and:
If x∈X then µ({x})=a,
If H K ∈C(X) then µ(H)⊳µ(K).
Let L = ω ×[0,1)∪{ω } with the following order topology, where we use the
1 1
symbol < for the usual orders on the sets ω and [0,1):
1
ω = min(L)
1
(α,r) ⊳ (β,s) if α>β
(α,r) ⊳ (β,s) if α=β and r >s.
We callL withthe ⊳orderthe invertedlongarc. Fornotationalconveniencewe
define (α,1)=(α+1,0).
Theorem 5.1. Let M = lim{M ,fβ} be a non-metric one-dimensional
ω1 ←− α α α<β<ω1
continuum constructed in Section 3. Then there is a generalized Whitney map
µ:C(Mω1)→L.
Since the one-dimensional continua M can be decomposed into hereditarily
ω1
indecomposablecontinuatoyieldametricone-dimensionalcontinua,thetechniques
of Hern´andez-Guti´errez [12] can be used to construct the Whitney map. For the
sake of completion, we provide an outline of a construction consistent with our
approach. Wewillneedthe followingfactaboutcontinuousdecompositionsofone-
dimensional metric continua as constructed above (see [22]). Suppose that M is a
one-dimensional metric continuum, X is a one-dimensional metric continuum and
G is the continuous decomposition of X into pseudo-arcs so that M = X/G. Let
H ⊂ X. If H intersects two elements of G, then H = ∪{g ∈ G|H ∩g 6= ∅}. In
particular we have the following property:
Property D: If H is a subcontinuum of X, then H is either contained in a single
element of the decomposition, or it is a union of decomposition elements.
Notation: Giventheabove,wheref :X →M istheopenmonotonemapassociated
with the decomposition space, if H ⊂M then f(H)=∪{f(x)|x∈H}.
Lemma 5.2. Suppose that M is a one-dimensional metric continuum, X is a one-
dimensional metric continuum and G is the continuous decomposition of X into
10 JANP.BORON´SKIANDMICHELSMITH
pseudo-arcs so that M = X/G. Suppose that µ : C(M) → [0,1] is a Whitney
M
map with µ (M) = 1. Then there exists a Whitney map µ : C(X) → [0,2] so
M X
that µ (f−1(H))=µ (H)+1 for all H ∈C(M).
X M
Proof. Let ν : C(X)→[0,2] be an onto Whitney map. If K is a subcontinuum of
some element of G then let g denote that element; for K ∈C(X) define
K
ν(K) if K ⊂g
µX(K)= ν(gK) K
( µM(f(K))+1 if K *g,∀g ∈G
Sinceforeachg ∈G,ν(g)6=0,thefunction ν(K) iswelldefinedandcontinuous.
ν(gK)
Furthermore, if K =g for some g ∈G we have
ν(K) ν(g )
K
= =1.
ν(g ) ν(g )
K K
From the definition of ν and property D, it follows if H,K ∈ C(X) and H ( K
then µ (H)<µ (K). (cid:3)
X X
Now we are ready to prove Theorem 5.1.
Proof. LetM beanarbitraryonedimensionalmetriccontinuum. Forα>0,G =
0 α
{fα−−11)p)| p∈Xα−1} is the continuous decompositionof Mα into pseudo-arcs used
to construct M . For each α let µ :C(M )→[0,1] be a Whitney map. For each
α α α
successor ordinal there is a Whitney map µ :M →[0,1] so that µ (f−1 (K))=
α α α α−1
1+µα(K)foreachK ∈C(Mα−1). We now constructµ:C(Mω1)→Linductively.
Foreachα<ω letJ denote the setofsubcontinua ofM so thatH ∈J if and
1 α c ω1 c α
only if α is the first ordinal such that π (H) is nondegenerate. Then J ∩J = ∅
α α β
if α 6= β and C(M ) = ∪ J ∪{{x}|x ∈ M }. Let µ : C(M ) → [0,1] be a
ω1 α<ω1 α ω1 0 0
Whitney map. Suppose H ∈J then define:
0
µ(H)=(0,1−µ (H)).
0
Observe that if H and K are two elements of J and H ( K then µ(H)⊳µ(K);
0
alsoobserve that the continuity ofµ implies that µ| is continuous. Also observe
0 J0
that µ(Mω1) is the maximal element of L with respect to the order ⊳. Suppose
now that α is such that µ(H) has been constructed for all H ∈J with β <α.
β
Case 1: α is a limit ordinal. Observe that for limit ordinals J =∅.
α
Case 2: α is a successor ordinal and H ∈ J . Then µ(K) had been defined
α
for K ∈ ∪ J so that µ(K) ∈ [(α,1),(0,0)] and restricted to ∪ J , µ has
β<α β β<α β
the Whitney property. Since Mα−1 is a continuous decomposition of Mα into
pseudo-arcssatisfyingpropertyD,wewishtoapplythelemma. Sincetopologically
[(α,1),(0,0)]⊳ isequivalentto[0,1]and[(α−1,1),(α−1,0)]⊳ istopologically[1,2]
wecanletφ:[0,2]→[(α−1,1),(α,0)]be a homeomorphismsothatφ(1)=(α,1).
Then we apply the lemma to obtain µ(H) in terms of µ (H).
Mα
Finally for x ∈ M define µ({x}) = ω . If H ∈ J , K ∈ J and α < β
ω1 1 α β
then H * K and µ(K)⊳µ(H). If H,K ∈ Jα and H ⊂ K then by construction
µ(H)⊳µ(K). SothefactthatµhastheWhitneypropertycanbeeasilyverifiedfrom
the fact that at each stage of the construction the Whitney property is preserved.
We wish to verify the continuity of µ: First observe that for H ∈ C(M ), if α
ω1
is the first ordinal so that π (H) is not degenerate then α is not a limit ordinal.
α
So (α,1)⊳µ(H) E (α,0) = (α−1,1); then for some t < 1 there is an open set U
in C(M ) so that if K ∈U then (α,0)⊳µ(K)⊳(α−1,t). So µ is continuous at
ω1
H. Consider x ∈ M and let α < ω , observe that ∪ J ∪{{x}|x ∈ M } is
ω1 1 β>α β ω1