A Core Decomposition of Compact Sets in the Plane
Benoît Loridant∗1 , Jun Luo†2 , and Yi Yang3
1
2,3
Montanuniversität Leoben, Franz Josefstrasse 18, Leoben 8700, Austria
School of Mathematics, Sun Yat-Sen University, Guangzhou 512075, China
3
Corresponding author: [email protected]
July 14, 2017
Abstract
A compact metric space is called a generalized Peano space if all its components are
locally connected and if for any constant C > 0 all but finitely many of the components
are of diameter less than C. Given a compact set K ⊂ C, there usually exist several
upper semi-continuous decompositions of K into subcontinua such that the quotient space,
equipped with the quotient topology, is a generalized Peano space. We show that one of
these decompositions is finer than all the others and call it the core decomposition of K
with Peano quotient. For specific choices of K, this core decomposition coincides with two
models obtained recently, namely the locally connected models for unshielded planar continua
(like connected Julia sets of polynomials) and the finitely Suslinian models for unshielded
planar compact sets (like disconnected Julia sets of polynomials). We further answer several
questions posed by Curry in 2010. In particular, we can exclude the existence of a rational
function whose Julia set is connected and does not have a finest locally connected model.
Keywords. Locally connected, finitely Suslinian, core decomposition.
∗
This author was supported by the Agence Nationale de la Recherche (ANR) and the Austrian Science Fund
(FWF) through the project FAN ANR-FWF I1136.
†
This author was supported by the Chinese National Natural Science Foundation Projects 10971233 and
11171123 .
1
1
Introduction and main results
In this paper, a compact metric space is called a compactum and a connected compactum is
called a continuum. If K, L are two compacta, a continuous onto map π : K → L such that the
preimage of every point in L is connected is called monotone [21].
We are interested in compacta and continua in the plane or in the Riemann sphere. Given
a compactum K ⊂ C, an upper semi-continuous decomposition D of K is a partition of K such
that for every open set B ⊂ K the union of all D ∈ D with D ⊂ B is open in K (see [13]).
Let π be the natural projection sending x ∈ K to the unique element of D containing x. Then
a set A ⊂ D is said to be open in D if and only if π −1 (A) is open in K. This defines the
quotient topology on D. An upper semi-continuous decomposition of K is monotone if each of
its elements is a subcontinuum of K. In this case, π is a monotone map. Finally, let D and D′ be
two monotone decompositions of a compactum K ⊂ C, with projections π and π ′ , and suppose
that D and D′ both satisfy a topological property (T ). We say that D is finer than D′ with
respect to (T ) if there is a map g : D → D′ such that π ′ = g ◦ π. If a monotone decomposition
of a compactum K is finer than every other monotone decomposition of K with respect to (T ),
T
then it is called the core decomposition of K with respect to (T ) and it will be denoted by DK
or simply DK , if (T ) is fixed. Clearly, the core decomposition DK is unique, if it exists.
Recently, core decompositions with respect to two specific topological properties were studied
for the special class of compacta K ⊂ C that are unshielded, that is, K = ∂W for the unbounded
component W of C \ K. More generally, when dealing with subsets of the Riemann sphere, a
compactum K ⊂ Ĉ is called unshielded if K = ∂W for some component W of Ĉ \ K. In
particular, if a rational function R : Ĉ → Ĉ has a completely invariant Fatou component, then
its Julia set is an unshielded compactum (see [1, Theorem 5.2.1.(i)]).
LC of
First, Blokh-Curry-Oversteegen prove in [4, Theorem 1] that the core decomposition DK
an unshielded continuum K with respect to the property of being locally connected always exists.
A special case is when K is the connected Julia set of a polynomial. In our paper, we will solve the
LC for all continua K ⊂ C, without assuming that K is unshielded. In particular,
existence of DK
our core decomposition will apply to the study of connected Julia sets of rational functions on
the extended complex plane Ĉ, thus negatively answers [6, Question 5.2]. However, if we only
consider upper semi-continuous decompositions then there might be two decompositions D1 , D2
of an unshielded continuum K ⊂ C which are both Peano continua under quotient topology,
such that the only decomposition finer than D1 and D2 is the decomposition {{z} : z ∈ K}
into singletons. Actually, let C ⊂ [0, 1] ⊂ C be Cantor’s ternary set. Let K be the union of
2
{x + iy : x ∈ C, y ∈ [0, 1]} with {x + i : x ∈ [0, 1]} and call it the Cantor Comb. See the following
figure for an approximation of K. Let p1 be the restriction to K of the projection x + iy 7→ x.
O
Figure 1: A rough approximation of the Cantor Comb.
n
o
LC
Then D1 = p−1
1 (x) : x ∈ [0, 1] coincides with DK , the core decomposition of K with respect
to local connectedness. Let D2 be the union of all the translates Cy := {x + iy : x ∈ C} of K
with 0 ≤ y ≤ 1 and all the single point sets {z = x + i} with x ∈
/ C. Then D2 is an upper
semi-continuous decomposition of K, which is not monotone and which is a Hawaiian earing
under quotient topology. Clearly, the only decomposition finer than both D1 and D2 is the
decomposition {{x} : x ∈ K} into singletons.
Second, given an unshielded compactum K ⊂ C, the result of [2, Theorem 4] indicates the
existence of the core decomposition of K with respect to the property of being finitely Suslinian,
F S . Here we recall that a compactum is finitely Suslinian if every collection of
denoted by DK
pairwise disjoint subcontinua whose diameters are bounded away from zero is finite. Since every
finitely Suslinian continuum is locally connected, we see that [4, Theorem 1] is a special case
F S of an
of [2, Theorem 4]. We may wonder about the existence of the core decomposition DK
arbitrary compactum K ⊂ C. However, there are examples of continua K ⊂ C failing to have
such a core decomposition (see [2, Example 14] and Section 2 of this paper). We will replace
the property of being finitely Suslinian by the property of being a generalized Peano space.
This class of compacta will be defined below. For every compactum K ⊂ C, we will prove the
P S with respect to the property of being a generalized
existence of the core decomposition DK
P S the core decomposition of K with Peano quotient. Since a
Peano space. We will briefly call DK
P S is finer
finitely Suslinian compactum is also a generalized Peano space, the decomposition DK
F S for any compactum K ⊂ C, when the latter exists. If the compactum K ⊂ C is
than DK
P S coincides with the finest
unshielded, we will prove in Section 2 that our core decomposition DK
F S , given by [2, Theorem 4].
finitely Suslinian model DK
LC we will obtain just coincides with D P S , when K ⊂ C is a
The core decomposition DK
K
continuum. It is unknown whether such a core decomposition exists for a general continuum or
3
compactum K, e.g., when K can not embedded into the plane.
Our work is motivated by recent studies and possible applications in the field of complex
dynamics, but we will rather focus on the topological part. The concept of generalized Peano
space has its origin in an ancient result by Schönflies. It also has motivations from some recent
works by Blokh, Oversteegen and their colleagues. We will show that this property can be
used advantageously in discussing core decompositions, besides the properties of being locally
connected or finitely Suslinian. Schönflies’ result reads as follows.
Theorem. [13, p.515, §61, II, Theorem 10]. If K is a locally connected compactum in the plane
and if the sequence R1 , R2 , . . . of components of C \ K is infinite, then the sequence of their
diameters converges to zero.
The above theorem gives a necessary condition for planar compacta to be locally connected.
This condition is also necessary for planar compacta to be finitely Suslinian, as we will prove in
Theorem 4.1. However, in both cases, the condition is not sufficient. For instance, Sierpinski’s
universal curve is not finitely Suslinian but its complement has infinitely many components whose
diameters converge to zero. Also, the closed topologist’s sine curve is not locally connected but
its complement has a single component. This motivates us to introduce the following condition,
which happens to be necessarily fulfilled by every planar compactum K, if K is assumed to be
finitely Suslinian or locally connected.
Schönflies Condition. A compactum K in the plane fulfills the Schönflies condition if for the
region U bounded by any two parallel lines L1 and L2 , the difference U \ K has at most finitely
many components intersecting both L1 and L2 .
Theorem 1. If a compactum K in the plane is locally connected or finitely Suslinian then it
satisfies the Schönflies condition.
We will prove an equivalent formulation of the Schönflies condition in Lemma 3.3: a compactum K ⊂ C satisfies the Schönflies condition if and only if for the region U bounded by any
two parallel lines L1 and L2 , the intersection U ∩K has at most finitely many components that
intersect both L1 and L2 . Moreover, we will show that the above Schönflies condition entirely
characterizes the local connectedness for continua in the plane.
Theorem 2. A continuum K in the plane is locally connected if and only if it satisfies the
Schönflies condition.
Continua like Sierpinski’s universal curve indicate that the above theorem does not hold if
we replace locally connected with finitely Suslinian.
4
The main purpose of this paper is not to characterize the finitely Suslinian compacta, but to
find appropriate candidates for the core decomposition, instead of the finitely Suslinian property.
Such a core decomposition of planar compacta will have interesting applications to the study
on Julia sets of rational functions. See for instance the open questions proposed at the end of
[6]. Combined with earlier models developed by Blokh-Curry-Oversteegen [2, 4], the results of
Theorems 1 and 2 provide some evidence that planar compacta satisfying the Schönflies condition
seem to be a reasonable model for the above mentioned core decompositions. Therefore, we give
a nontrivial characterization of the Schönflies condition as follows.
Theorem 3. A compactum K ⊂ C satisfies the Schönflies condition if and only if it has the
following two properties.
(1) Every component of K is locally connected.
(2) For every C > 0, all but finitely many components of K are of diameter less than C.
A compact metric space satisfying the above two properties in Theorem 3 will be called a
generalized Peano space, simply Peano space. In particular, a Peano space is a Peano continuum
if it is connected. Note that the term “Peano space” has been used as a synonym for Peano
continuum in the literature (see [8, p.199] or [9, p.117]). In our paper, a connected Peano space
means a Peano continuum, while a Peano space might be disconnected.
P S , the core
We are now in the position to introduce our strategy to prove the existence of DK
decomposition with Peano quotient, for every compactum K in the plane.
Let us define a relation RK on any given compactum K ⊂ C as follows. Given two disjoint
simple closed curves J1 and J2 , we denote by U (J1 , J2 ) the component of Ĉ \ (J1 ∪ J2 ) bounded
by J1 ∪ J2 . This is an annulus in the extended complex plane Ĉ. We say that two points
x, y ∈ K are related under RK provided that there exist two disjoint simple closed curves J1 ∋ x
and J2 ∋ y such that U (J1 , J2 ) ∩ K contains an infinite sequence of components Qk intersecting
both J1 and J2 , whose limit lim Qk under Hausdorff distance contains {x, y}.
k→∞
Definition 4. Let K ⊂ C be a compactum and RK the relation defined above. We denote by
RK be the collection of all the closed equivalence relations on K containing RK (as subsets of
K × K). Moreover, we denote by ∼ the intersection of all the equivalence relations of RK . It is
also an element of RK , and we call it the minimal equivalence containing RK , or the Schönflies
equivalence on K, for short.
One can check that the equivalence class [x] under the Schönflies equivalence ∼ is a continuum
for every x ∈ K (see Proposition 5.1). Moreover, if K in Theorem 3 is assumed to be unshielded,
5
then it is finitely Suslinian if and only if it satisfies the Schönflies condition (see Theorem 2.1).
Also, if the compactum K in Definition 4 is unshielded, then ∼ coincides with the relation
developed in [2], given by the finest finitely Suslinian model (see Section 2).
From now on, we denote by DK the collection of equivalence classes [x] := {z ∈ K : z ∼ x}.
Thus DK is the decomposition induced by the Schönflies equivalence on K. It is standard to
verify that DK is necessarily a compact, Hausdorff and secondly countable space under quotient
topology [11, p.148, Theorem 20]. Therefore, it is metrizable by Urysohn’s metrization theorem
[11, p.125, Theorem 16]. We will prove that it satisfies the two properties of Theorem 3, thus is
a Peano space.
Theorem 5. Under quotient topology DK is a Peano space.
After this we will show that DK is finest in the following sense.
Theorem 6. Let ∼ be the Schönflies equivalence on a compactum K ⊂ C and π(x) = [x] the
natural projection from K to DK . If f : K → Y is monotone map onto a Peano space Y , then
there is an onto map g : DK → Y with f = g ◦ π.
P S of K
By Theorems 5 and 6, we can conclude that DK equals the core decomposition DK
with Peano quotient.
P S with respect to the
Theorem 7. Every compactum K ⊂ C has a core decomposition DK
property of being a Peano space. It coincides with the decomposition DK induced by the Schönflies
equivalence ∼ on K.
Remark 8. Theorem 7 answers [6, Question 5.2] and partially answers [6, Question 5.2]. In the
first part of [6, Question 5.2], Curry asks: for what useful topological properties P does there
exist a finest decomposition of every Julia set J(R) (of a rational function R) satisfying P? By
Theorem 7, the property of “being a Peano space” is such a property. Moreover, in the last part
of [6, Question 5.2], Curry asks: which of these (properties) is the appropriate analogue for the
P S in Theorem 7 generalizes the
finest locally connected model? Since the core decomposition DK
earlier finest models obtained in [2, 4], the answer is again the property of “being a Peano space”.
Moreover, in the middle part of [6, Question 5.2], Curry asks: is the decomposition (satisfying
the right property P) dynamic? This interesting question provides a reasonable angle to apply
the core decomposition obtained in Theorem 7 to the study of complex dynamics, in particular,
to the study on dynamics of a rational function restricted to its Julia set.
Remark 9. For any compactum K, the decomposition {Q : Q is a component of K} always
induces a Peano quotient, whose components are single points. Therefore, an important problem
6
P S of a given compactum K ⊂ C induces a
is to determine whether the core decomposition DK
P S is a nonquotient space having a non-degenerate component. In such a case, we say that DK
degenerate core decomposition; otherwise, we say that it is a degenerate core decomposition.
Clearly, the core decomposition of every indecomposable continuum K ⊂ C is degenerate, since
P S = {K}. The studies of Blokh-Curry-Oversteegen [2, 4], whose models are generalized by
DK
the core decomposition introduced in this paper, already provide very interesting results on
the existence of non-degenerate core decompositions. For instance, by [4, Theorem 27], if a
continuum X ⊂ K has a “well-slicing family”, then the image of X under the natural projection
P S is a non-degenerate continuum, hence K has a non-degenerate core decomposition.
π : K → DK
If K is the Julia set of a polynomial, then it is stated in [2, Corollary 24] that K has a nonP S if and only if K has a periodic component Q which, as a
degenerate core decomposition DK
P S . In other words, to compute the
plane continuum, has a non-degenerate core decomposition DQ
P S we just need to compute the core decomposition D P S for all the periodic
core decomposition DK
Q
components Q of K. If the above Julia set K is connected and is “finitely irreducible”, the result
P S satisfies either D P S = {K} or
of [6, Theorem 4.1] indicates that the core decomposition DK
K
P S = {{x} : x ∈ K}. Actually, in the former case K is an indecomposable continuum and in
DK
the latter case it is homeomorphic to [0, 1]. Finally, if X is an unshielded continuum and Y ⊂ X
is a subcontinuum, Blokh-Oversteegen-Timorin [5] obtained recently a sufficient condition for
the core decomposition DYP S of Y to embed canonically into that of X. As an application to
complex dynamics, the authors also considered the special case that X is the connected Julia
set of a renormalizable polynomial P and Y is the so-called small Julia set, for a polynomiallike map obtained as a restriction of some iterate P n with n > 1. Combining these results
with the core decomposition obtained in our paper, one may investigate problems like the local
connectedness of Julia set of infinitely renormalizable polynomials.
We arrange our paper as follows. Section 2 briefly recalls facts on local connectedness,
laminations in complex dynamics and core decompositions. We provide an argument based
P S and D F S of an unshielded compactum
on Theorems 3 and 5 that the core decompositions DK
K
K ⊂ C are equal. Section 3 gives preliminary lemmas needed in the proofs of the main theorems.
Section 4 proves Theorems 1 to 3. Sections 5 and 6 respectively prove Theorems 5 and 6.
Acknowledgments. We are grateful to the referee for pointing out significant and confusing
typos and for an interesting question that led to Remark 9.
7
2
Local Connectedness, Lamination, and Core Decomposition
The investigation of local connectedness dates back to the nineteenth century. Cantor proved
that the unit interval and the unit square have the same cardinality. In other words, there
exists a bijection h : [0, 1] → [0, 1]2 , and this map h can not be continuous. Peano and some of
his contemporaries further obtained continuous surjections from [0, 1] onto planar domains like
squares and triangles. The range of a continuous map from [0, 1] into a metric space is therefore
often called a Peano continuum. Peano continua were then fully characterized via the notion
of local connectedness: indeed, Hahn and Mazurkiewicz showed that a continuum is a Peano
continuum if and only if it is locally connected.
Among the Peano continua of the plane, the boundary of a bounded simply connected domain
U provides a special case. By the Riemann Mapping Theorem, there is a conformal isomorphism
from the unit open disk D = {|z| < 1} onto U . Furthermore, Carathéodory’s theorem states
that this conformal mapping has a continuous extension to the closed disk D if and only if the
boundary ∂U is locally connected. Considering U as a domain in the extended complex plane
Ĉ, we may assume, after the action of a Möbius map, that ∞ ∈ U . Then X = C \ U is a full
continuum, i.e., it has a connected complement U . Moreover, there is a conformal isomorphism
Φ from D∗ = {|z| > 1} ∪ {∞} onto U , fixing ∞ ∈ Ĉ and having a real derivative at ∞.
In the study of quadratic dynamics, examples of the above map Φ are (1) Böttcher maps for
hyperbolic polynomials z 7→ z 2 + c with c lying in a hyperbolic component of the Mandelbrot
set M and (2) the conformal isomorphism sending D∗ onto Ĉ \ M.
For the map Φ in (1), the boundary of Φ(D∗ ) is the Julia set Jc of z 7→ z 2 + c, which is
known to be locally connected. In this case, Jc is the image of the unit circle ∂D = ∂D∗ under
a continuous map (called Carathéodory’s loop), hence may be considered as the quotient space
of an equivalence relation on ∂D. This equivalence relation is a lamination in Thurston’s sense
[20]. Douady [7] proposed a pinched disc model describing full locally connected continua in
the plane. Extending the lamination in a natural way to a closed equivalence relation L on the
closed unit disk D, he obtains that Kc is homeomorphic with the quotient D/L, where Kc is the
filled Julia set of the polynomial z 7→ z 2 + c (Jc = ∂Kc ).
The pinched disc model works even if the full continuum K is not locally connected. The
map Φ in (2) provides a typical example, in which the boundary of Φ(D∗ ) coincides with that
of the Mandelbrot set M. Denote by Rθ the image of {re2πθi : r > 1} under Φ for θ ∈ [0, 1]. Rθ
is called the external ray at θ. If lim Φ re2πθi is a point on ∂M, denoted as cθ , we say that
r→1
Rθ lands at cθ . It is known that all external rays Rθ with rational θ lands. Douady therefore
8
n
o
[7] defines an equivalence relation on e2πθi : θ ∈ Q ∩ [0, 1]
′
by setting θ ∼Q
M θ if and only if
cθ = cθ′ . As a subset of ∂D×∂D, the closure of ∼Q
M (denoted ∼M ) turns out to be an equivalence
relation on ∂D (see [7, Theorem 3] for fundamental properties of ∼M ).
Let us now recall the main ideas of Blokh-Curry-Oversteegen [4] concerning locally connected
models for unshielded continua in the plane. Let K ⊂ Ĉ be an unshielded continuum with
K = ∂U , where U is the unbounded component of C \ K. Let Φ be a conformal mapping that
sends D∗ to U and fixes ∞. For any θ ∈ [0, 1], the impression at e2πθi , defined by
Imp(θ) =
∗
2πθi
lim Φ(zi ) : {zi } ⊂ D , lim zi = e
i→∞
i→∞
,
is a subcontinuum of K. By [4, Lemma 13] there is a minimal closed equivalence relation I on
K such that every equivalence class is made up of impressions and is a subcontinuum of K. By
[4, Lemma 16], if R is an arbitrary closed equivalence on K such that the quotient space K/R
is a locally connected continuum then I is contained in R (as subsets of K × K). The first part
of [4, Lemma 17] obtains that the quotient K/I is a locally connected continuum, called the
locally connected model of K. Now we may define a closed equivalence relation ∼K on ∂D by
requiring that θ ∼K θ′ if and only if Imp(θ) and Imp(θ′ ) lie in the same equivalence class [x]I .
Then, by the second part of [4, Lemma 17], the equivalence ∼K is a lamination such that the
induced quotient ∂D/ ∼K is homeomorphic to K/I.
In particular, when K is the Julia set of a polynomial f of degree d ≥ 2 without irrationally
neutral cycles, Kiwi [12] investigates the structure of the classes [x]I and shows that every [x]I
coincides with the fiber at x ∈ K [12, Definition 2.5] defined by
Fiber(x) = {y ∈ K : no finite set separates y from x in K} .
Here, a finite set C ⊂ K separates two points of K if these points are in distinct components
of K \ C. We refer to [12, Corollary 3.14] and [12, Proposition 3.15] for important properties of
Fiber(x), and to Schleicher’s earlier works [17, 18, 19] for another approach in defining fibers.
To generalize the above model, Blokh, Curry and Oversteegen [2] define an equivalence ≃
on an unshielded compactum K ⊂ C to be the minimal closed equivalence such that every
limit continuum is contained in a single class [x]≃ := {z ∈ K : z ≃ x}. Recall that a limit
continuum is the limit lim Nk under Hausdorff distance of an infinite sequence of pairwise
k→∞
F S = {[x]
disjoint subcontinua Nk ⊂ K. The quotient space DK
≃ : x ∈ K} is necessarily a
compact metrizable space [16, p.38, Theorem 3.9]. The authors of [2] further check that it is
finitely Suslinian [2, Lemma 13].
F S , as a subset of C, possesses the following
Every element d of the above decomposition DK
9
property: the union of all the bounded components of C \ d does not intersect K. The authors
F S is the finest monotone decomposition of K
of [2] then use Moore’s theorem to prove that DK
F S is the core decomposition
with finitely Suslinian quotient [2, Theorem 19]. In other words, DK
of K with respect to the finitely Suslinian property.
≃ = {[x] : x ∈ K} with ≃ defined as above. On
Let now K ⊂ C be a any compactum. Let DK
≃
the other side, let ∼ be the Schönflies equivalence on K, defined in Definition 4 as the minimal
closed equivalence relation containing the relation RK . We write DK = {[x]∼ : x ∈ K}. We
want to compare these two decompositions. The definition of RK indicates that if (z1 , z2 ) ∈ RK
then there is a limit continuum containing both z1 and z2 . This in turn indicates that ∼ is
≃.
contained in ≃ as subsets of K × K, hence the decomposition DK always refines DK
These two decompositions turn out to be equal provide that K is unshielded. Note that
≃ = D F S is the core decomposition of K with respect to the finitely Suslinian
in this case DK
K
property [2, Theorem 19]. Actually, the unshielded assumption of K implies that the bounded
components of C \ d for every d ∈ DK are all disjoint from K. Let d∗ be the union of d with the
bounded components of C \ d. Then
DC := {d∗ : d ∈ DK } ∪



{z} : z ∈
/

[
d∈DK


d∗ 

is a monotone decomposition of C, such that d∗1 ∩ d∗2 = ∅ for any d1 6= d2 ∈ DK . By Moore’s
Theorem, the quotient DC is homeomorphic to the plane and the natural projection Π : C → DC
sends K to a planar compactum. Since every d∗ is disjoint from the unbounded component W
of C \ K, the image Π(W ) is a region in the plane DC whose boundary contains Π(K). That
is to say, Π(K) is also an unshielded compactum in the plane DC . On the other hand, for any
x, y ∈ K it is direct to check that Π(x) = Π(y) if and only if π(x) = π(y). Therefore, the
quotient DK is homeomorphic to Π(K) hence may be embedded into the plane as an unshielded
compactum. Theorem 5 of this paper says that DK is also a Peano space. By the following
theorem, such a planar compactum is finitely Suslinian. Consequently, the core decomposition
F S is finer than D , and we have D F S = D .
decomposition DK
K
K
K
Theorem 2.1. If an unshielded compactum K ⊂ C is a Peano space then it is finitely Suslinian.
Proof. By Theorem 3 and the definition of Peano space, we only need to consider the case when
K is an unshielded continuum. Recall that a continuum X is regular at a point x ∈ X if for every
neighborhood Vx of x there exists a neighborhood Ux of x whose boundary ∂Ux = Ux ∩ X \ Ux is
a finite set [22, p.19]. A regular continuum is just one that is regular at each of its points. Here
it is standard to check that a regular continuum is finitely Suslinian. Therefore, our proof will
10
be completed if only we can verify that K, a locally connected unshielded planar continuum, is
a regular continuum.
We will use the notions of pseudo fiber and fiber for planar continua, recently introduced
in [10], from which a numerical scale is developed that measures the extent to which such a
continuum is locally connected.
More precisely, for any point x ∈ K, the pseudo fiber Ex at x consists of the points y ∈ K
such that there does not exist a simple closed curve γ with γ ∩ ∂X a finite set, called a good
cut, such that x and y lie in different component of C \ γ; the fiber Fx at x is the component
of Ex that contains x. By [10, Proposition 4.2], for the locally connected unshielded continuum
K ⊂ C every pseudo fiber Ex equals the single point set {x}. Therefore, given any x ∈ K and
any open set U ∋ x, we can choose good cut γy such that x and y lie in different component of
C \ γy . Let Uy , Vy be the component of C \ γy with x ∈ Uy and y ∈ Vy . Then {Vy : y ∈ K \ U }
is an open cover of the compact set K \ U . Fix a finite sub-cover {Vy1 , . . . , Vyn }. Then
Ux :=
n
\
U yi
i=1
is open in K, contains x, and is contained in U . Recall that, for 1 ≤ i ≤ n, the intersection
Bi := Uyi ∩ Vyi ∩ K is contained in γyi ∩ K hence is also a finite set. Since the boundary of Ux
in K is defined to be the intersection Ux ∩ K ∩ (K \ Ux ) and is a subset of
which is in turn a subset of
S
i Bi
n [
Ux ∩ K ∩ V yi ∩ K ,
i=1
and hence is also a finite set. This verifies that K is regular
at x. Consequently, from flexibility of x ∈ K we can infer that K is a regular continuum.
Note that another proof of this theorem can be found in [3, Lemma 2.7].
We mention that if the compactum K ⊂ C is not assumed to be unshielded, then “the” core
decomposition of K with respect to the finitely Suslinian property may not exist. Consider for
instance the locally connected continuum K ⊂ C of [2, Example 14]. It admits two monotone
decompositions D1 , D2 such that the quotients are finitely Suslinian. However, the only partition
finer than both D1 and D2 is the trivial decomposition {{x} : x ∈ K}. Therefore, “the” core
decomposition of K with respect to the finitely Suslinian property does not exist, while the
trivial decomposition {{x} : x ∈ K} is the core decomposition of K with respect to the property
of being a Peano space.
We end up this section with an example of a continuum K ⊂ C having two properties.
Firstly, the core decomposition DK has an element d such that at least two components of C \ d
intersect K; secondly, the resulted quotient space DK can not be embedded into the plane.
11
Example 2.2. Let the compactum K ⊂ C be the union of the closure of the unit disk D =
{z ∈ C : |z| < 1} and the spiral curve L =
n
o
1 + e−t e2πit : t ≥ 0 . By routine works one may
check that the core decomposition of K with respect to the property of being a Peano space is
exactly given by
DK = {{x} : x ∈ (D ∪ L)} ∪ {∂D}.
Clearly, the quotient space is the one-point union of a sphere with a segment, thus can not be
embedded into the plane.
3
Some Useful Lemmas
The lemmas in this section give a couple of results that are used in latter sections. Lemma 3.1
is from [14, Lemma 2.1] and will be used in proving Lemmas 3.2 and 3.3.
Lemma 3.1. Suppose that A ⊂ [0, 1) × [0, 1] and B ⊂ (0, 1] × [0, 1] are disjoint closed sets. Then
there exists a path in [0, 1]2 \ (A ∪ B) starting from a point in (0, 1) × {0} and leading to a point
in (0, 1) × {1}.
Before stating the next lemma, we recall the following definitions and facts. For X ⊂ C, we
say that X = A ∪ B (A, B 6= ∅) is a separation of X if A ∩ B = A ∩ B = ∅.
Remember that, if x0 is a point in X then the component of X containing x0 is the maximal
connected set P ⊂ X with x0 ∈ P . The quasi-component of X containing x0 is defined to be
the set
Q = {y ∈ X : no separation X = A ∪ B exists such that x ∈ A, y ∈ B}.
Equivalently, the quasi-component containing a point p ∈ X may be defined as the intersection of all closed-open subsets of X containing p. Any component is contained in a quasicomponent, and quasi-components coincide with the components whenever X is compact [13].
If X is compact we denote by X ∗ be the union of X with all the bounded components of
C \ X. We call X ∗ the topological hull of X, following Blokh-Curry-Oversteegen [2].
Lemma 3.2. Let K ⊂ C be compactum and x0 ∈
/ K. Then x0 lies in the unbounded component
of C \ K provided that it does not lie in the topological hull P ∗ for any component P of K.
Proof. Fix a large enough circular disk Dr with radius r > 0 whose interior contains K. For
each component P of K, since x0 is assumed to be in the unbounded component of C \ P , we
may choose a path αP disjoint from P which joins x0 to a fixed point x1 ∈ ∂Dr . Let δ be a
12
number smaller than the distance dist(P, αP ) := {|z1 − z2 | : z1 ∈ P, z2 ∈ αP }. As P is also
a quasi-component of K, we may choose a separation K = Ay,P ∪ By,P with P ⊂ Ay,P and
y ∈ By,P for any point y ∈ K with dist(y, P ) ≥ δ. By compactness of {y ∈ K : dist(y, P ) ≥ δ},
there are finitely many points y1 , . . . , yl ∈ K with dist(yi , P ) ≥ δ such that
By1 ,P , By2 ,P , . . . , Byl ,P
are open under the induced topology of K and form a cover of {y ∈ K : dist(y, P ) ≥ δ}. Clearly,
K = AP ∪ BP is also a separation with αP ∩ AP = ∅, where
AP =
l
\
Ayi ,P
and BP =
l
[
Byi ,P .
i=1
i=1
Since every AP is both open and closed under the induced topology of K, we see that there are
finitely many components P1 , . . . , Pm with
K=
m
[
i=1
!
APi .
Rename αPi as αi for 1 ≤ i ≤ m. Let A1 = AP1 . And, for 2 ≤ i ≤ m, let

Ai = APi \ 
i−1
[
j=1


Aj  = APi \ 
Then A1 , . . . , Am are disjoint compact sets such that K ⊂
i−1
[
j=1
Sm

APj  .
i=1 Ai .
Moreover, every αi is an arc
satisfying αi ∩ Ai = ∅. To finish our proof, we will show that there is a path α disjoint from K
that joins x0 to x1 .
Recall that the paths α1 , α2 : [0, 1] → C, with common initial point αi (0) = x0 and common
endpoint αi (1) = x1 , are homotopic relative to {0, 1} under the straight line homotopy F :
[0, 1]2 → C defined by F (t, s) = tα1 (s) + (1 − t)α2 (s) for any t, s ∈ [0, 1]. The disjointness
of A1 , A2 indicates that F −1 (A1 ), F −1 (A2 ) are disjoint compact subsets of the unit square. By
Lemma 3.1 there is a path β in [0, 1]2 \ F −1 (A1 ) ∪ F −1 (A2 ) starting from a point in {0} × (0, 1)
and leading to a point in {1} × (0, 1). Then F (β) is a path disjoint from A1 ∪ A2 and joins x0
to x1 .
Repeating the above argument on the paths F (β) and α3 , we can find a path disjoint from
A1 ∪ A2 ∪ A3 that joins x0 to x1 . Inductively, we can find a path α disjoint from ∪m
i=1 Ai , hence
disjoint from K, that joins x0 to x1 .
Lemma 3.3. Let K ⊂ C be a compact set and U the region bounded by two parallel lines L1
and L2 . Let Ui be the component of C \ (L1 ∪ L2 ) with ∂Ui = Li . We have the following:
(1) If K ∩ L1 6= ∅ =
6 K ∩ L2 and no connected subset of K intersects both L1 and L2 , then there
is a separation K = A1 ∪ A2 with A1 , A2 compact sets such that (Ui ∩ K) ⊂ Ai .
13
(2) If U \ K has at least m ≥ 2 components intersecting both L1 and L2 , then U ∩ K has at least
m − 1 components intersecting both L1 and L2 .
(3) If U ∩ K has at least m ≥ 2 components intersecting both L1 and L2 , then U \ K has at least
m components intersecting both L1 and L2 .
Remark 3.4. The results of Lemma 3.3 still hold if we replace C by Ĉ, L1 , L2 by two disjoint
simple closed curves J1 , J2 ⊂ Ĉ and U by the region W bounded by J1 ∪ J2 . For Part (1), the
argument is still valid. For Part (2), we may remove an open arc α ⊂ W , which joins a point in
J1 to a point in J2 and has a closure disjoint from K; then the difference W \ α contains W ∩ K
and has the same topology of U . For part (3), one may use the pattern of polar brick wall tiling,
as indicated in the figure below, to find such an open arc α.
r=1
R=2
Figure 2: A Polar Brick Wall Tiling, in which every two tiles either are disjoint or intersect at
a non-degenerate arc on the boundary.
Proof for Lemma 3.3. We first prove Item (1). Every component of the compact set U ∩ K
is also a quasi-component. Therefore, for any a ∈ (L1 ∩ K) and any b ∈ (L2 ∩ K), there exists a
separation U ∩ K = Wa,b ∪ Va,b with a ∈ Wa,b and b ∈ Va,b such that the sets Wa,b , Va,b are closed
in C and relatively open in U ∩ K. In particular, the collection {Va,b : b ∈ (L2 ∩ K)} is an open
cover of the compact set L2 ∩ K in U ∩ K, so there exists a finite subcover Va,b1 , ..., Va,bk . Let
Wa =
k
\
Wa,bi
and Va =
k
[
Va,bi .
i=1
i=1
Then U ∩ K = Wa ∪ Va is a separation such that a ∈ Wa and (L2 ∩ K) ⊂ Va . By flexibility of
a ∈ L1 ∩ K, the collection {Wa : a ∈ (L1 ∩ K)} is an open cover of L1 ∩ K in U ∩ K and has a
finite subcover Wa1 , . . . , Wal . Let
W =
l
[
W ai
and V =
l
\
i=1
i=1
14
Va i .
Then U ∩ K = W ∪ V is a separation such that (L1 ∩ K) ⊂ W and (L2 ∩ K) ⊂ V . If we set
A1 = W ∪(U1 ∩K) and A2 = V ∪(U2 ∩K), then K = A1 ∪A2 is a separation with (U1 ∩K) ⊂ A1
and (U2 ∩ K) ⊂ A2 . By construction, A1 and A2 are closed subsets of C. This proves Item (1).
The proof of Item (2) reads as follows. Let R1 , . . . , Rm be m components of U \K intersecting
both L1 and L2 . For each i ∈ {1, . . . , m}, let αi be a simple arc in the component Ri of U \ K
with endpoints ai ∈ L1 , bi ∈ L2 and αi \ {ai , bi } ⊂ U ∩ Ri . Choose a line L perpendicular to L1
such that all of the arcs {αi }16i6m are in the same component of C \ L. We can assume that a1
is the nearest point to L among the collection {ai }16i6m , and aj is the nearest point to L among
the set of points {ai }j6i6m , then the arcs αi are renamed according to the distance of their
endpoints on L1 to the line L. Since the arcs αi are disjoint, we can use Jordan curve theorem
to infer that the distance from bj to L is smaller than that from bj+1 to L for 1 ≤ j ≤ m − 1.
Let βi be the arc on L1 with endpoints ai , ai+1 for 1 ≤ i ≤ m − 1. Let γi be the arc on
L2 with endpoints bi , bi+1 for 1 ≤ i ≤ m − 1. Then Γi = αi ∪ βi ∪ γi ∪ αi+1 is a simple closed
curve for 1 ≤ i ≤ m − 1. Let Wi ⊂ U be the bounded component of C \ Γi . By a theorem of
Schönflies [15, p.68,Theorem 6], there is a homeomorphism hi : Wi → [0, 1]2 such that
hi (ai ) = (0, 0),
hi (ai+1 ) = (0, 1),
hi (bi ) = (1, 0),
hi (bi+1 ) = (1, 1).
Since αi and αi+1 lie in distinct components of U \ K, the compact set hi Wi ∩ K intersects
each of hi (βi ) and hi (γi ) and is disjoint from hi (αi ∪ αi+1 ) = {0, 1} × [0, 1]. See left part of
Figure 3 for relative locations of hi (αi ), hi (αi+1 ), hi (βi ) and hi (γi ).
hi (ai+1 )
hi (αi+1 )
hi (βi )
hi (bi+1 )
hi (αi+1 )
hi (ai )
hi (bi )
hi (αi )
L∗2 U2∗
U1∗ L∗1
hi (γi )
hi (αi )
Figure 3: Relative locations of the points hi (ai ), hi (bi ), hi (ai+1 ), hi (bi+1 ) in [0, 1]2 .
We claim that hi Wi ∩ K has a component Pi that intersects both hi (βi ) and h(γi ). This
will verify Item (2), since Pi is a closed subset of [0, 1]2 , hence Ni := h−1
i (Pi ) ⊂
Wi ∩ K
is a sub-continuum of K and intersects both L1 and L2 . Note that Ni ∩ Ni+1 = ∅, since
Wi ∩ Wi+1 ∩ K = αi+1 ∩ K = ∅.
15
Suppose on the contrary that hi Wi ∩ K has no component intersecting both hi (βi ) and
h(γi ). We will use Item (1) to induce a contradiction.
To this end, we put K ∗ = hi Wi ∩ K . Then, let L∗1 be the line through hi (ai ), hi (ai+1 ) and
L∗2 the line through hi (bi ), hi (bi+1 ). Moreover, let Ui∗ be the component of C \ (L∗1 ∪ L∗2 ) with
∂Ui∗ = L∗i . See right part of Figure 3.
By Item (1), we are able to find a separation K ∗ = A∗1 ∪A∗2 into compact sets with (Ui∗ ∩K ∗ ) ⊂
A∗i . It follows that A∗1 ⊂ [0, 1)×[0, 1] and A∗2 ⊂ (0, 1]×[0, 1]. In particular, A∗1 and A∗2 satisfy the
conditions in Lemma 3.1, from which we can infer the existence of a path P in [0, 1]2 \ (A∗1 ∪ A∗2 )
starting at a point in (0, 1) × {0} ⊂ hi (αi ) and leading to a point in (0, 1) × {1} ⊂ hi (αi+1 ).
The inverse h−1
i (P ) is then a path in Wi \ K (⊂ U \ K), which connects a point in αi ⊂ Ri to a
point in αi+1 ⊂ Ri+1 . This is impossible, since Ri , Ri+1 are distinct components of U \ K.
Finally we prove part (3). Let Q1 , . . . , Qm be m components of U ∩ K intersecting both L1
and L2 . Clearly ǫ =
1
3
min {dist(Qi , Qj ) : 1 ≤ i < j ≤ m} is a positive number. For 1 ≤ i ≤ m,
let Qi (ǫ) be the open ǫ-neighborhood of Qi . Then for every point x in (U ∩ K) \ Qi (ǫ) the
quasi-component of U ∩ K containing x is disjoint from the one containing Qi , so that there is a
separation U ∩ K = Cx ∪ Dx with Qi ⊂ Cx and x ∈ Dx . Under the induced topology of U ∩ K,
the collection
o
n
Dx : x ∈ (U ∩ K) \ Qi (ǫ)
is an open cover of the compact set (U ∩ K) \ Qi (ǫ), which has a finite sub-cover
n
o
Dxk : x1 , · · · , xn ∈ (U ∩ K) \ Qi (ǫ) .
Then U ∩ K = Ci ∪ Di is a separation with Qi ⊂ Ci ⊂ Qi (ǫ) for 1 ≤ i ≤ m, where
Ci =
n
\
Cxk
and Di =
k=1
n
[
Dxk .
k=1
Moreover, U ∩ K = C ∪ D is also a separation, where
C=
m
[
Ci
and D =
m
\
Di .
i=1
i=1
Denote by d the distance between L1 and L2 . Let δ = min{dist(C, D), ǫ}. Fix an integer N1 > 0
with
d
2N1
<
δ
4
and divide U into 2N1 equal strips by 2N1 − 1 lines parallel to L1 such that the
width of each strip is
d
2N1 .
In the rest part of our proof we assume that L1 is the horizontal axis and L2 is on the upper
half plane.
16
Let T1 = {Bk : k ∈ Z} be a tiling of the lowest strip by squares of side length
d
2N1 ,
such that
T1 is a cover and the squares have disjoint interiors. Then
T2n−1 := Bk +
(2n − 2)d
,0 : k ∈ Z
4N1
is a tiling of the (2n − 1)-th strip for 2 ≤ n ≤ N1 , and
T2n := Bk +
(2n − 1)d d
,
4N1
4N1
:k∈Z
is a tiling of the (2n)-th strip for 1 ≤ n ≤ N1 . Moreover, T :=
2N
[1
Ti is a tiling of the whole strip
i=1
U that represents a “brick wall pattern”, such that two squares either are disjoint or intersect
at a non-degenerate segment.
For 1 ≤ i ≤ m, let Ci′ be the union of all the squares in T intersecting Ci . Let Ri be the
component of Ci′ containing Qi . Clearly, we have Qi ⊂ Ci ⊂ Ci′ ⊂ Qi (ǫ). By Torhorst Theorem
[13, p.512, §61, II, Theorem 4], the unbounded component of C \ Ri is bounded by a simple
closed curve Ji . Clearly, the curves J1 , . . . , Jm are pairwise disjoint.
Choose ai ∈ (Ji ∩ L1 ) and bi ∈ (Ji ∩ L2 ) with dist(ai , L) = dist(Ji ∩ L1 , L) and dist(bi , L) =
dist(Ji ∩ L2 , L). Then Ji \ {ai , bi } is made up of two open arcs; one of them must be contained
in U and will be denoted as Ai . Let H be the union of L1 , L2 and all the arcs A1 , . . . , Am . Fix
a permutation i1 i2 · · · im of 1, 2, . . . , m such that the distance from aik to L is smaller than that
from aik+1 to L for 1 ≤ k ≤ m − 1. Then C \ H has exactly m − 1 components W1 , W2 , . . . , Wm−1
and the boundary of Wk is Aik ∪ Aik+1 ∪ αk ∪ βk for 1 ≤ k ≤ m, where αk ⊂ L1 and βk ⊂ L2
are minimal arcs containing aik , aik+1 and bik , bik+1 , respectively.
Since Jik ∩ Jik+1 = ∅, from the choice of the points aik , bik and the arc Aik , we can infer that
Jik \ Aik is contained in the closure Wik , which necessarily contains Qik . Since Qik intersects
both L1 and L2 , the arc Aik is separated from all the arcs Ail with l > k by Qik in U . That is
to say, all the arcs A1 , A2 , . . . , Am lie in different components of U \ K, indicating that U \ K
has m components which intersect both L1 and L2 .
Let RK be the closed relation on a compact set K ⊂ C, firstly mentioned before Definition 4.
The following lemma provides an equivalent approach to define RK . In the sequel, the bounded
component of a simple closed curve Γ ⊂ C is denoted as Int(Γ) and called the interior of Γ.
Lemma 3.5. Given a compact set K ⊂ C and n ≥ 3 disjoint simple closed curves Γ1 , . . . , Γn ⊂ C
such that Γ2 , . . . , Γn ⊂ Int(Γ1 ) and Int(Γj ) ∩ Int(Γj ) = ∅ for i 6= j ≥ 2. Let W be the
annulus bounded by Γ1 , Γ2 . Let W ∗ be the only region with ∂W ∗ =
17
Sn
k=1 Γk .
If the intersection
W ∗ ∩ K has infinitely many components Pk each of which intersects both Γ1 and Γ2 , such
that limk Pk = P∞ under Hausdorff distance, then (z1 , z2 ) ∈ RK for any z1 ∈ (Γ1 ∩ P∞ ) and
z2 ∈ (Γ2 ∩ P∞ ).
Proof. We only consider the case n = 3, to which the other cases may be reduced.
Fix a circular disk D whose interior contains K. By Lemma 3.2, there are two mutually
exclusive possibilities: (1) the interior Int(Γ3 ) is contained in P ∗ for some component P of
W ∗ ∩ K or (2) a point x0 ∈ Γ3 may be connected to a point x1 on the circle ∂D by a path α
disjoint from W ∗ ∩ K.
In the former case, every Pk other than P is disjoint from P ∗ ; each of those components
is a component of W ∩ (K ∪ P ∗ ) hence also a component of W ∩ K. This guarantees that
(z1 , z2 ) ∈ RK for any z1 ∈ (Γ1 ∩ P∞ ) and z2 ∈ (Γ2 ∩ P∞ ).
In the latter case, let y be the first point on α that also lies in Γ1 ∪ Γ2 and α0 ⊂ α the
irreducible sub-path with end points x0 , y. There are two subcases, y ∈ Γ2 or y ∈ Γ1 , and we
just consider the subcase y ∈ Γ2 since the same argument applies to the other subcase.
We may slightly thicken α0 and find two disjoint arcs α′ , α′′ close enough to α0 , each of which
does not intersect W ∗ ∩K and joins a point on Γ3 to a point on Γ2 , such that α′ , α′′ are contained
in W ∗ except for their end points. Then the unbounded component of C \ (Γ2 ∪ Γ3 ∪ α′ ∪ α′′ )
is bounded by a simple closed curve Γ′2 contained in Int(Γ2 ), such that the region W ′ bounded
by Γ1 , Γ′2 is an annulus. Clearly, we have W ∗ ∩ K = W ′ ∩ K, and thus every Pk is also a
component of W ′ ∩ K. Consequently, we have (z1 , z2 ) ∈ RK for any z1 ∈ (Γ1 ∩ P∞ ) and
z2 ∈ (Γ2 ∩ P∞ ) ⊂ (Γ′2 ∩ P∞ ).
4
Proofs for Theorems 1 to 3
Firstly, we copy the ideas of Schönflies result [13, p.515, §61, II, Theorem 10] and obtain a
necessary condition for a planar compactum to be finitely Suslinian.
Theorem 4.1. Given a finitely Suslinian compactum K ⊂ C. If the sequence R1 , R2 , . . . of
components of C \ K is infinite then the sequence of their diameters converges to zero.
Proof. Suppose conversely that there exists ǫ > 0 and infinitely many integers i1 < i2 < · · ·
such that the diameter δ(Rin ) > 3ǫ. For each component Rin , choose an arc αin ⊂ Rin with
diameter larger than 3ǫ. We may assume that αin converges to α0 under Hausdorff distance.
18
Here we have δ(α0 ) ≥ 3ǫ. Choose two points p′ , q ′ ∈ α0 with |p′ − q ′ | = 3ǫ. Then, we can fix two
points p1 , p2 in the interior of the segment pq with |p1 − p2 | = 2ǫ.
Let Li be the line through pi which is perpendicular to pq. Let U be the region bounded by
L1 ∪ L2 . Since lim αin = α0 , there exists an integer N such that αin intersects both L1 and
n→∞
L2 for all n > N . By part (2) of Lemma 3.3, there exist infinitely many components of U ∩ K
which intersect both L1 and L2 . This contradicts the condition that K is finitely Suslinian.
Secondly, we prove Theorem 1 as follows.
Proof for Theorem 1. The part for locally connected compacta is a direct corollary of Schönflies’ result [13, p.515, §61, II, Theorem 10]. So we only consider the part for finitely Suslinian
compacta.
Suppose on the contrary that there exist two parallel lines L1 , L2 such that the difference
U \ K has infinitely many components R1 , R2 , . . . intersecting each of L1 and L2 . Here U is the
only component of C \ (L1 ∪ L2 ) bounded by L1 ∪ L2 . By part (2) of Lemma 3.3, U ∩ K has
infinitely many components intersecting each of L1 and L2 . This contradicts the assumption
that K is finitely Suslinian.
Then we continue to prove Theorem 2.
Proof for Theorem 2. We just show that a continuum K ⊂ C satisfying the Schönflies condition is locally connected. Suppose on the contrary that K is not locally connected at a point
x0 ∈ K. By definition of local connectedness [13, p.227, §49, I, Definition], there would exist
a closed square V centered at x0 such that the component P0 of V ∩ K containing x0 is not
a neighborhood of x0 with respect to the induced topology on V ∩ K. In other words, there
exists a sequence {xk }∞
k=1 in (V ∩ K) \ P0 with lim xk = x0 such that the components of V ∩ K
k→∞
containing xk , denoted Pk , are pairwise disjoint.
Recall that the hyperspace of all closed nonempty subsets of V is a compact metric space
under Hausdorff distance. Coming to an appropriate subsequence, if necessary, we may assume
that Pk converges to P∞ in Hausdorff distance. From this, we see that P∞ is a sub-continuum
of P0 and that the diameter of Pk , denoted δ(Pk ), converges to δ(P∞ ).
By connectedness of K, each Pk must intersect ∂V hence P∞ intersects ∂V . Since Pk → P∞
under Hausdorff distance, we can pick some point y0 ∈ (∂V ∩ P∞ ) and points yk ∈ (∂V ∩ Pk )
for all k ≥ 1 such that y0 := lim yk .
k→∞
19
Since ∂V consists of four segments and contains the infinite set of points {yk },we may fix a
line L1 crossing infinitely many yk , which necessarily contains y0 . Then, fix a line L2 parallel to
L1 which separates x0 from y0 , so that x0 and y0 lie in different components of C \ L2 . Let U be
the strip bounded by L1 and L2 . Obviously, there exists an integer N such that Pn intersects
both L1 and L2 for n > N . Without loss of generality, we may assume that every Pk intersects
both L1 and L2 .
It follows that for all k ≥ 1 the intersection U ∩ Pk = V ∩ U ∩ Pk has a component Qk
intersecting both L1 and L2 . Otherwise, by part (1) of Lemma 3.3 there is a separation U ∩Pk =
A1 ∪ A2 , where Ai is the union of all the components of U ∩ Pk intersecting Li . Let U1 , U2 be
the two components of C \ U , with Li = ∂Ui . Then Pk = [A1 ∪ (U1 ∩ Pk )] ∪ [B1 ∪ (U2 ∩ Pk )] is
a separation. This contradicts connectedness of Pk
Therefore, by part (3) of Lemma 3.3, our proof will be completed if only we can show that
for all but two integers k ≥ 1 the continuum Qk is also a component of U ∩ K. To this end,
for all k ≥ 1 we may fix two points ak ∈ (L1 ∩ Qk ) and bk ∈ (L2 ∩ Qk ). Then, we fix a line L
perpendicular to L1 and disjoint from K. See Figure 4. Now, we call a continuum Qk a nearest
L1
L
Lj,1
Lk,1
ak
aj
Qj
L2
al
Qk
Ql
bk
bj
Lj,2
Ll,1
Lk,2
bl
Ll,2
Figure 4: Relative locations of ak , bk and Qj , Qk , Ql .
component (respectively a furthest component) if
dist (ak , L) < max {dist (aj , L) : j 6= k}
(dist (ak , L) > max {dist (aj , L) : j 6= k}) .
Clearly, there exist at most one nearest component and at most one furthest component. We
claim that all the other Qk is also a component of U ∩ K. Actually, if Ui denotes the component
of C \ U with ∂Ui = Li we can choose for all k ≥ 1 two rays Lk,1 ⊂ U1 and Lk,2 ⊂ U2 parallel to
L such that ak ∈ Lk,1 and bk ∈ Lk,2 . The above Figure 4 gives a simplified depiction for relative
locations of L, Li and aj , ak , al .
If Qk is neither nearest nor furthest there exist two components Qj , Ql with dist (aj , L) <
dist (ak , L) < dist (al , L). In this case, we can use an appropriate brick wall tiling of U and the
Jordan curve theorem to infer that Qk \ (L1 ∪ L2 ) is contained in a bounded component W of
20
C \ M , where M = aj al ∪ bj bl ∪ Qj ∪ Ql is a continuum which lies entirely in V . Therefore,
(W ∪ M ) ∩ K is a subset of V ∩ K; and we are able to choose a separation W ∩ K = A ∪ B with
Qk ⊂ A and (Qj ∪ Ql ) ∩ A = ∅. From this we can infer that Qk is also a component of A, which
is disjoint from B1 := (U \ W ) ∩ K. That is to say, the intersection U ∩ K is divided into two
disjoint compact subsets, A and B ∪ B1 . Combining this with the fact that Qk is a component
of A, we already verify that Qk is also a component of U ∩ K.
Finally, we prove Theorem 3.
Proof of Theorem 3. Suppose that (1) and (2) hold and assume that K does not satisfy the
Schönflies condition. Then there exists a region U bounded by two parallel lines L1 and L2 such
that U ∩ K has infinitely many components {Nk } which intersect both L1 and L2 . Due to (2),
there exists infinitely many {Nki } in one component of K which is denoted by P , and these
Nki are also in different components of U ∩ P . That is to say, P does not satisfy the Schönflies
condition. By Theorem 2, we reach a contradiction to the local connectedness of P . This verifies
the “if” part.
To prove the “only if” part we assume that K satisfies the Schönflies condition and verify
conditions (1) and (2) as follows.
Given any component P of K and the region U bounded by any two parallel lines L1 and
L2 , it is routine to check that every component of U ∩ P is also a component of U ∩ K. Thus
U ∩ P has finitely many components intersecting both L1 and L2 . By Theorem 2, P is locally
connected.
If (2) is not true, so that there exist an infinite sequence of sub-continua {Nk } lying in distinct
components of K, denoted Qk , such that their diameters δ(Nk ) are greater than a positive
constant C, then we can choose a subsequence {Nkn } converging to a continuum N∞ ⊂ X
under Hausdorff distance. Clearly, the diameter δ(N∞ ) ≥ C. So we can choose two points
x, y ∈ N∞ with |x − y| = C and two parallel lines L1 , L2 perpendicular to the line crossing
x, y and intersecting the interior of the segment xy. Now we can see that all but finitely many
Nk ⊂ Qk must intersect L1 and L2 at the same time. This implies that for all but finitely many
integers k ≥ 1, there exists a component Pk of U ∩ Qk intersecting both L1 and L2 . Here U is
the region bounded by L1 , L2 . For those integers k, the continuum Pk is also a component of
U ∩ K, which contradicts the assumption that K satisfies the Schönflies condition.
21
5
The Quotient DK is a Peano Space
Throughout this section, we fix a compact set K in the plane and denote by ∼ the Schönflies
equivalence relation given in Definition 4; moreover, the decomposition DK of K is made up of
the equivalence classes [x] = {y ∈ K : x ∼ y}, for x ∈ K. Then DK is an upper semi-continuous
decomposition. In the following proposition we further show that its elements are all connected,
so that the natural projection π : K → DK is a monotone map.
Proposition 5.1. Every class [x] of ∼ is a continuum.
Proof. Assume that there exist two disjoint simple closed curves J1 ∋ x and J2 ∋ y such that
the region W bounded by J1 and J2 has the following property: the common part W ∩ K has
infinitely many components P1 , P2 , . . . that intersect both J1 and J2 and converge to a continuum
P∞ under Hausdorff distance. We only need to verify that for any point z ∈ P∞ and any small
number r > 0 there is a point y ′ with |z − y ′ | < r such that y ′ is related to x under the relation
RK given before Definition 4. This then implies the inclusion P∞ ⊂ [x]. By flexibility of x and
y and by minimality of ∼ we will obtain that every class of ∼ is connected.
Considering K and W as subsets of the sphere and applying a Möbius transformation, if
necessary, we may assume with no loss of generality that W is an open annulus. Then we may
consider it as the standard annulus {1 < |z| < 2}. Let W be covered with a polar brick wall
tiling, as indicated in the left part of Figure 5. Fix a separation W ∩ K = A ∪ B with P1 ⊂ A
x ∈ γ1 ⊂ J1
Pk
α1
z
β1
y ∈ γ2 ⊂ J2
Figure 5: Polar Brick Wall Tiling and relative locations of Pk and z ∈ W .
and P∞ ⊂ B. Assume that every tile in the polar brick wall tiling is of diameter less than
1
3 dist(A, B),
where dist(A, B) := inf{|a − b| : a ∈ A, b ∈ B} is the distance between A and B.
Let A∗ be the union of all the tiles intersecting A, and B ∗ the union of those intersecting B.
Then each of A∗ and B ∗ is a finite disjoint union of Peano continua. By the construction of
polar brick wall tiling, each of those Peano continua has no cut point.
22
Let M be the component of A∗ that contains P1 . By Torhorst Theorem [13, p.512, §61,
II, Theorem 4], the unbounded component of C \ M is bounded by a simple closed curve Γ.
Moreover, the curve Γ has exactly two arcs α1 , β1 each of which connects a point on γ1 to a
point on γ2 . Then it is clear that W \ (α1 ∪ β1 ) is the union of two regions, each of which is
bounded by a simple closed curve. One of these two regions contains P1 and the other contains
P∞ . The latter is denoted D1′ and depicted in the right part of Figure 5. Clearly, the boundary
of D1′ is the simple closed curve J1′ = α1 ∪ β1 ∪ γ1 ∪ γ2 .
Now, for any point z ∈ (P∞ ∩ D1′ ) and any number r > 0 that is smaller than dist(z, J1′ ), we
may fix a small open circular disk D2′ ⊂ D1′ centered at z with radius r and denote its boundary
as J2′ . Recall that infinitely many Pk will intersect D2′ . For such a Pk the difference Pk \ D2′ has
a component Qk intersecting both J1′ , J2′ . Since those Qk are also components U ′ ∩ K, where
U ′ = D1′ \ D2′ , and since we may choose an appropriate subsequence of {Qk } that is convergent
under Hausdorff distance, we already show that (x, y ′ ) ∈ RK for some y ′ ∈ J2′ , which obviously
satisfies the inequality |z − y ′ | < r.
Now we assume that DK is equipped with a metric d, which is compatible with the quotient
topology. To prove Theorem 5, we start from a special case when the compactum K is a
continuum.
Theorem 5.2. If K is a continuum then DK is locally connected under quotient topology.
Proof. If DK is not locally connected at some point π(x), where x is a point in K and π(x) = [x],
then there exists a closed ball Bε in DK centered at π(x) with radius ε > 0 whose component
containing the point π(x) is not a neighborhood of π(x). Let Q be the component of Bε
containing π(x). Then we can find an infinite sequence π(xk ) in Bε \ Q with d(π(xk ), π(x)) → 0
such that for k 6= l the components Qk ∋ π(xk ) and Ql ∋ π(xl ) are disjoint.
Fix a point π(yk ) on Qk ∩ ∂Bε for all k ≥ 1. By coming to an appropriate subsequence,
if necessary, we may assume xk → x∞ and yk → y∞ . Then, continuity of π guarantees that
π(x∞ ) = lim π(xk ) = π(x) and that π(y∞ ) = lim π(yk ) belongs to the boundary of Bε .
k→∞
k→∞
Now, considered as subsets of K, we see that π(x∞ ) = [x∞ ] and π(y∞ ) = [y∞ ] are disjoint
planar continua. In particular, we have [x∞ ] ⊂ π −1 (Bεo ) and [y∞ ] ⊂ E := π −1 [(DK ) \ Bεo ] =
K \ π −1 (Bεo ), where Bεo denotes the interior of Bε . Since π : K → DK is a monotone map, the
pre-images π −1 (Q), π −1 (Qk ) are disjoint sub-continua of K. Moreover, there exist two disjoint
open sets V1 and V2 satisfying:
[x∞ ] ⊂ V1 ⊂ π −1 (Bεo ) ,
[y∞ ] ⊂ E ⊂ V2 ,
23
π(V1 ) ∩ π(V2 ) = ∅.
(1)
By compactness of K we may assume that K is contained in a strip {x + iy : x ∈ R, 0 ≤ y ≤ 1},
which will be covered by rectangles of the form
n
Rk,j
:=
(
2k + 2 1 − (−1)j
j
j+1
2k 1 − (−1)j
≤
x
≤
+
, n ≤y≤ n
x + iy : n +
n+1
n
n+1
2
2
2
2
2
2
n
n
with n ≥ 1, k ∈ Z and 0 ≤ j ≤ 2n − 1. Then Rk,j
o
)
.
(2)
n is called
is a tiling of R × [0, 1]; every Rk,j
a tile and every two tiles either are disjoint or intersect at a non-degenerate segment. Let Tn be
n that intersects E, and S the union of those that intersect [x ].
the union of all the tiles Rk,j
n
∞
Then each of Tn , Sn is the union of finitely many rectangles and Sn is connected. Clearly, they
have no cut points; moreover, the interior of Tn contains E and that of Sn contains [x∞ ].
Choose a large integer n ≥ 1 satisfying Sn ⊂ V1 and Tn ⊂ V2 , so that π(Tn ) ∩ π(Sn ) = ∅.
Let W1 be the component of C \ Sn containing [y∞ ]. Since Sn is locally connected and has no
cut point, by Torhorst Theorem [21, p.126] we see that the boundary J1 := ∂W1 is a simple
closed curve. Moreover, the difference W1 \ Tn is an open set (possibly not connected) bounded
by two or more disjoint simple closed curves. It has a unique component W ∗ whose boundary
contains J1 and all the other components of W1 \ Tn have a positive distance to J1 . Since the
equivalence class [y∞ ] is a continuum lying in W1 , the boundary of W ∗ contains at least another
simple closed curve J2 , which separates J1 from [y∞ ], and consists of finitely many disjoint
simple closed curves, say J1 , J2 , . . . , Jm . Since J1 ⊂ Sn and (J2 ∪ J3 ∪ · · · ∪ Jn ) ⊂ Tn , we have
π(J1 ) ∩ π(J2 ∪ · · · ∪ Jm ) = ∅.
(3)
The rest of our proof is to obtain a contradiction to this equation. To this end, we firstly
recall that, for every large enough k > 1, the pre-image π −1 (Qk ) is a continuum intersecting
both J1 and J2 . Then it follows that W ∗ ∩ π −1 (Qk ) has a component Pk intersecting both J1
and some Ji with 2 ≤ i ≤ m. Let 2 ≤ i0 ≤ m be an integer such that that Pk ∩ Ji0 6= ∅ for
infinitely many k.
We may assume that J2 ⊂ Int(J1 ), by applying a Möbius transformation if necessary.
Since W ∗ ∩ E = ∅, we have W ∗ ∩ K ⊂ π −1 (Bε−ξ ) for a small ξ ∈ (0, ε). Combing this
with the fact that the continuum π −1 (Qk ) is a component of π −1 (Bε ), we can infer that Pk is
also a component of W ∗ ∩ K. Choose a subsequence of {Pk } that converges to a limit continuum
P∞ under Hausdorff distance. Then P∞ intersects J1 and Ji0 at the same time. By Lemma 3.5,
we have (z1 , z2 ) ∈ RK for all z1 ∈ (J1 ∩ P∞ ) and z2 ∈ (Ji0 ∩ P∞ ) ⊂ (J2 ∪ · · · ∪ Jm ). This is
impossible by Equation (3).
Then, we discuss the case when K is a disconnected compactum.
24
Proof for Theorem 5. By Theorem 5.2, we just need to show that for any ε > 0 there are at
most finitely many components of DK which are of diameter greater than ε.
Otherwise, there is an infinite sequence {Qj : j} of components whose diameters are greater
than a constant ε0 > 0. By monotonicity of the natural projection π : K → DK , the preimages
Pj := π −1 (Qj ) are components of K. By uniform continuity of π, the diameters of Pi are
greater than a constant t0 > 0. By going to a subsequence, if necessary,we may assume that the
sequence {Pj } converges to a limit continuum P∞ under Hausdorff distance. Then, continuity
of π ensures that Qj → π(P∞ ), which is a continuum of diameter ≥ ε0 . So, we can fix two
points x1 , x2 ∈ P∞ with d(π(x1 ), π(x2 )) ≥ ε0 . Clearly, the two equivalence classes [x1 ], [x2 ] are
disjoint continua in the plane.
Since DK = {[z] : z ∈ K} is an upper semi-continuous decomposition, we may fix two open
sets U1 ⊃ [x1 ] and U2 ⊃ [x2 ] with disjoint closures such that [z] ∩ U2 = ∅ for all z ∈ U1 .
Let Ji be the circle centered at xi with an arbitrary radius r > 0 such that Ji ⊂ Ui . Let W
be the component of C \ (J1 ∪ J2 ) with ∂W = J1 ∪ J2 . The containment {x1 , x2 } ⊂ P∞ implies
that all but finitely many Pj intersect both J1 and J2 . Recall that every Pj is a component of K.
For each of those Pj intersecting both J1 and J2 , the intersection W ∩ K has a component Mj ,
which intersects J1 and J2 both. Those Mj are each a component of a component of W ∩ Pj .
Therefore, a subsequence of {Mj } must converge to a limit continuum M∞ , which intersects
both J1 and J2 . Fixing any point z ∈ M∞ ∩ J1 , we have z ∈ U1 and ([z] ∩ U2 ) ⊃ ([z] ∩ J2 ) 6= ∅.
This contradicts the fact mentioned in the previous paragraph.
6
DK is a Core Decomposition
This section has a single aim, to prove Theorem 6. And we only need to show that f (x1 ) =
f (x2 ) for any x1 , x2 ∈ K with (x1 , x2 ) ∈ RK , since ∼ is the smallest closed equivalence on K
containing RK and since {f −1 (y) : y ∈ Y } is also a monotone decomposition of K.
Proof for Theorem 6. By Definition 4, we may fix two disjoint simple closed curves Ji ∋ xi
such that W ∩ K has infinitely many components Pk , each of which intersects both J1 and J2 ,
such that the sequence {Pk } converges to a continuum P∞ ⊃ {x1 , x2 } under Hausdorff distance.
Here W is the only component of C \ (J1 ∪ J2 ) with ∂W = J1 ∪ J2 .
By applying an appropriate Möbius transformation, if necessary, we may assume with no loss
of generality that J1 separates J2 from infinity. By Schönflies Theorem (see [15, p.71, Theorem
25
3] or [15, p.72, Theorem 4]), we may consider W to be the annulus A = {z : 1 ≤ |z| ≤ 2}.
In the rest of this section, we denote by ρ the metric on Y and assume on the contrary that
f (x1 ) 6= f (x2 ). Then we could find a positive number ε0 < 12 ρ(f (x1 ), f (x2 )) such that the two
disks B1 , B2 ⊂ Y centered at x1 , x2 with radius ε0 have disjoint closures, i.e. B1 ∩ B2 = ∅. Let
Ui = f −1 (Bi ) for i = 1, 2. Then
U1 ∩ U2 = ∅ = f U 1 ∩ f U2 .
(4)
Since lim f (Pk ) = f (P∞ ) under Hausdorff distance, all but finitely many f (Pk ) are of diameter
k→∞
greater than 2ε0 . Since Y is a Peano space, all but finitely many f (Pk ) must be entirely contained
in a single component of Y , denoted Y0 , which also contains f (P∞ ). With no loss of generality
we may assume that every f (Pk ) is entirely contained in Y0 .
To complete our proof, we will induce a contradiction to local connectedness of Y0 , by showing
that Y0 is not locally connected at some point on f (P∞ ). More precisely, let K0 = K \ (U1 ∪ U2 );
then we will find a point y # ∈ f (P∞ ) ⊂ Y0 such that f (K0 ) is a neighborhood of y # and that
the component Q0 of f (K0 ) containing y # is not a neighborhood of y # . This clearly indicates
that the component of f (K0 ) ∩ Y0 containing y # is not a neighborhood of y # .
For every k ≥ 1, we can choose a separation A ∩ K = Ek ∪ Fk such that Pk ⊂ Ek , P∞ ⊂ Fk ,
and Pj ⊂ Fk for j 6= k. Cover the annulus A by a polar brick wall tiling Tk , whose tiles are all
of diameter strictly smaller than 12 dist(Ek , Fk ). Let Pk∗ be the component of
[
{T ∈ Tk : T ∩ Ek 6= ∅}
containing Pk . Then the unbounded component Wk of C \ Pk∗ contains P∞ and every Pj with
j 6= k. Since Pk∗ is a continuum with no cut point, by Torhorst Theorem [21, p.126], the
boundary of Wk is a simple closed curve Γk . Moreover, Γk contains exactly two sub-arcs αk , βk
each of which intersects both C1 := {|z| = 1} and C2 := {|z| = 2} and is otherwise contained
in the open annulus Ao . Let Dk be the component of C \ (C1 ∪ C2 ∪ αk ∪ βk ) that contains Pk .
Then ∂Dk is a simple closed curve for all k ≥ 1.
By appropriately choosing the sizes of tiles in Tk , we may further assume that the topological
disks Dk are pairwise disjoint. Then P∞ is contained in the difference A \ D1 , whose closure is
also a topological disk, denoted as D∞ . Now consider D∞ as the unit square [0, 1]2 .
Recall that D∞ ∩ K has infinitely many components Pk that intersects the top and the
bottom of [0, 1]2 at the same time. Therefore, the limit P∞ of a subsequence of {Pk } intersects
each of the two segments [0, 1] × {0} and [0, 1] × {1}. Let P ∗ be the component of D∞ ∩ K that
contains P∞ .
26
Let the two components of [0, 1]2 \ P ∗ containing {0} × [0, 1] and {1} × [0, 1] be respectively
denoted as UL and UR . Then, one of UL and UR contains infinitely many components of D∞ ∩K.
Assume that UL contains an infinite subsequence of {Pk } and rename this subsequence as {Qk }.
Then all but finitely many Qk intersect the two segments [0, 1] × {0} and [0, 1] × {1} both.
Now, fix two disks B(xi , r) centered at xi with radius r > 0 satisfying B(xi , r) ⊂ Ui . Since
Qk → P∞ ⊂ P ∗ under Hausdorff distance, we may find two points xk,i in (0, 1)2 ∩ Qk for a large
k ≥ 1 such that both |xk,1 − x1 | and |xk,2 − x2 | are smaller than min
x1
n
1
2, r
o
. Then, the segments
xk,1
Qk
xk,2
fj
α
fj
β
x2
fj and the points aj , bj , cj , dj .
fj , β
Figure 6: Relative locations of the component Pk , the arcs α
γk,i := xk,i xi for i = 1, 2 are disjoint and both γk,1 and γk,2 intersect all but finitely many of the
topological disks Dj that are constructed by using the polar brick wall tiling Tj . In particular,
each of γk,1 , γk,2 intersects both αj and βj for all j greater than some integer N ≥ k. For each
of those j > N , let aj be the last point of γk,1 that leaves αj and bj the first point of γk,1 after
aj that lies on βj ; let cj be the last point of γk,2 that leaves αj and dj the first point of γk,2 after
fj ⊂ αj connecting aj to bj and the
cj that lies on βj . Then, the segments aj bj , cj dj , the arc α
fj ⊂ βj connecting cj to dj form a simple closed curve, denoted γj . Since the disks Dj are
arc β
disjoint, so are the disks ∆j ⊂ Dj that are bounded by γj . For all the integers j > N , we have
two facts that will be used before we end our proof.
1. For all j ≥ 1, the intersection K ∩ aj bj ∪ cj dj = K ∩ (∂∆j ) = K ∩ γj is contained in
B(x1 , r) ∪ B(x2 , r), which is a subset of U1 ∪ U2 = f −1 (B1 ) ∪ f −1 (B2 ).
2. The intersection ∆j ∩ K has a component that intersects the segments aj bj and cj dj at
the same time. Denote this component by Mj .
Since limj aj bj = x1 and limj cj dj = x2 , for every j > N we may choose a point zj ∈ Mj
such that the distance from f (zj ) to B1 equals that from f (zj ) to B2 . Choose a convergent
subsequence of {zj }, denoted as {vn }, that converges to a point v∞ ∈ P∞ . Then the distance
27
from y # = f (v∞ ) to B1 equals that from y # to B2 . This implies that y # lies in the interior of
f (K0 ) = Y \ (B1 ∪ B2 ), where
K0 = K \ (U1 ∪ U2 ) = K \ f −1 (B1 ) ∪ f −1 (B2 ) .
Note that {f (vn ) : n} is an infinite sequence of points in f (K0 ) converging to y # such that the
components of f (K0 ) containing f (vn ) are pairwise disjoint. This implies that the component
Q0 of f (K0 ) containing y # is not a neighborhood of y # and ends the proof.
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