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TOPOLOGY
PROCEEDINGS
Volume 31, No. 1, 2007
Pages 151-161
http://topology.auburn.edu/tp/
AN UNCOUNTABLE COLLECTION OF
MUTUALLY INCOMPARABLE PLANAR FANS
CARLOS ISLAS
Abstract. In this paper, we show an uncountable collection
of mutually incomparable fans in the plane.
1. Introduction
A continuum means a nonempty compact connected metric space
and a map means a continuous function. An arc means a space
homeomorphic to the closed unit interval [0, 1]. A continuum X is
arcwise connected provided each two points of X are contained in
some arc contained in X. A continuum X is unicoherent if A ∩ B is
connected for each subcontinua A and B such that A ∪ B = X. X
is hereditarily unicoherent if every subcontinuum of X is unicoherent. A dendroid is an arcwise connected hereditarily unicoherent
continuum. A fan is a dendroid with only one ramification point.
We say that two continua are comparable by continuous maps if one
of those continua can be mapped onto the other. Otherwise, the
continua are incomparable. In 1961, B. Knaster [2] asked for an
uncountable family of continuously incomparable dendroids. Recently, this question was answered independently in [3] by Piotr
Minc and in [4]. These dendroids are fans, but it is not clear if they
are planable. Minc presented his example during the Spring Topology Conference held in Greensboro, North Carolina, in March 2006
2000 Mathematics Subject Classification. Primary 54F50, 54F15.
Key words and phrases. dendroids, fans, incomparable, planar.
The author thanks CONACyT and DGEP for the financial support provided.
c
2007
Topology Proceedings.
151
152
C. ISLAS
and Andrew Lelek asked if there exists such an example with planar dendroids. The same question was posed again in a session of
the Hyperspace Seminar, conducted by Professor Alejandro Illanes
at the University of Mexico (UNAM). In this paper, we answer the
question by constructing an uncountable family of incomparable
fans in the plane. Our family of fans is a modification of the collection of Elsa continua constructed by Marwan M. Awartani in [1].
We construct a family {Fα : α ∈ C}, C ⊂ 2N , C uncountable and
every Fα is a fan such that if α 6= β and a, β ∈ C then Fα and Fβ
are not comparable.
In the second section of this paper, we recall the examples given
by Awartani and introduce some notations that will be used in the
construction of the new family of fans.
In the third section, we construct our examples and we obtain
the main result, Theorem 3.8, using a modification of the lemmas
given in [1].
2. Awartani’s examples
Definition 2.1. Let 2N denote the set of all sequences of zeros
and ones, and let {ai }, {bi } be two elements in 2N . Then
(1) {ai } is said to vertically dominate {bi } if ai ≥ bi eventually.
(2) {ai } is said to dominate {bi } if there exists an integer j0
such that ai ≥ bi+j0 eventually.
(3) {ai } and {bi } are called incomparable if neither of them
dominates the other.
The proofs of the following results are in [1].
∞
Lemma 2.2. Let {ai }, {bi } be two elements in 2N . If aij j=1 is
a tail of the sequence of zeros in {ai }, and if for some integer j0 ,
bij +j0 is eventually zero, then {ai } dominates {bi }.
Lemma 2.3. 2N contains continuum many elements no one of
which vertically dominates any other.
Lemma 2.4. 2N contains continuum many elements none of which
is eventually constant and no one of which dominates any other.
Let C ⊂ 2N , C uncountable, such that if α 6= β and a, β ∈ C, no
one dominates the other.
INCOMPARABLE PLANAR FANS
153
With each α = {αi } ∈ 2N , we associate an Awartani example
(as in Figure 1) Eα as a compactification of the ray Jα with an arc
as remainder.
Figure 1. Eα with α1 = 0, α2 = 0, α3 = 1, . . .
Notice that each 1 in a sequence shifts one local
minimum to the 23 level.
Now, we will describe Jα , as the graph of a piece wise linear function
from (0, 1] to [0, 1], with (1, 0) = a1 as an end point. We consider
every point in Jα by the first coordinate. Moreover if a and b are
two points in Jα , then [a, b] denotes the subarc of Jα from a to b.
Let mα and Mα denote the set of minima and the set of maxima
of Jα and let Vα = mα ∪ Mα .
∞
Let {xi }∞
i=1 and {yi }i=1 be two sequences in (0, 1] such that
lim xi = 0 = lim yi and for every i, yi+1 < xi+1 < yi < xi .
We will obtain Jα as the union of subarcs [xi , yi ] and [yi , xi+1 ],
in such a way that
1. if αi = 0, then [xi ,yi ] ∩ Vα has the following properties:
a) |[xi , yi ] ∩ Mα | = 2i and |[xi , yi ] ∩ mα | = 2i − 1.
154
C. ISLAS
b) If {Mi,j : 1 ≤ j ≤ 2i} is an enumeration from right to
left of the elements of the set [xi , yi ] ∩ Mα , then
π2 (Mi,2i ) = 1 and π2 (Mi,j ) = π2 (Mi,2i−j ) = j+2
j+3 ,
1 ≤ j ≤ i.
c) π2 ([xi , yi ] ∩ mα ) = 12 ;
2. if αi = 1, then [xi , yi ] ∩ Vα has the following properties:
a) |[xi , yi ] ∩ Mα | = 2i + 1 and |[xi , yi ] ∩ mα | = 2i.
b) If {Mi,j : 1 ≤ j ≤ 2i + 1} is an enumeration from right
to left of the elements of the set [xi , yi ] ∩ Mα , then
π2 (Mi,2i+1 ) = 1 and π2 (Mi,j ) = π2 (Mi,2i+1−j ) = j+2
j+3 ,
1 ≤ j ≤ i.
c) If {mi,j : 1 ≤ j ≤ 2i} is an enumeration from right to
left of the elements of the set [xi , yi ] ∩ mα , then
π2 (mi,j ) = 23 for i = j and π2 (mi,j ) = 12 for i 6= j;
3. for each i ∈ N, [yi , xi+1 ] ∩ Vα has the following properties:
a) |[xi , yi ] ∩ Mα | = 2i − 1 and |[xi , yi ] ∩ mα | = 2i.
b) π2 ([xi , yi ] ∩ Mα ) = 12 .
c) If {mi,j : 1 ≤ j ≤ 2i} is an enumeration from right to
left of the elements of the set [xi , yi ] ∩ mα , then
π2 (mi,2i ) = 0 and π2 (mi,j ) = π2 (mi,2i−j ) = 1 − j+2
j+3 ,
1 ≤ j ≤ i.
Mi,2i if αi = 0
Let bi =
and ai = mi,2i , i.e., bi and ai are
Mi,2i+1 if αi = 1
the points in Jα such that its second coordinate is one and zero,
respectively.
{Eα : α ∈ C}, is an uncountable family and every Eα is an Elsa
continuum such that if α 6= β and a, β ∈ C, then Eα and Eβ are
not comparable.
3. Construction
Let α ∈ 2N and Eα be the Awartani example associated with α.
Though the fan we are interested in is Fα , we will construct a
fan Fα′ as an intermediate step with its only purpose being that it
is easily visualizable.
We will construct a fan Fα as a union of arcs Jiα . We start with
′
the construction of a fan Fα , as in the Figure 2, then we identify
INCOMPARABLE PLANAR FANS
155
the harmonic fan drawn in the inferior half of this figure with a
point aα to obtain the fan Fα .
W3
W2
U3
Figure 2. Fα′
W1
U2
U1
with α1 = 0, α2 = 1, α3 = 1, . . .
Now we will describe
Fα′ . Let us denote the points in the plane:
1
(0, 1) = cα , 0, 2 = bα , (0, 0) = aα , and (0, −1) = pα . We define
the arc Ei as the segment between pα and the point in the plane
1
,0 .
2i−1
′
If Ai = [ai , ai+1 ] is the arc between ai and ai+1 contained
in Jα ,
3
α
′
let I1 be a copy of A1 contained in the square 4 , 1 × [0, 1]. (It
means a contraction in the direction
of the first coordinate of the
graph Jα in [a2 , a1 ] × [0, 1] to 34 , 1 × [0, 1].) Then let I2α be the
union of two copies of A′2 , contained in the square 233 , 12 × [0, 1],
in such a way that the first copy is denoted by A2 and the second
copy is denoted by A1,2 . In general, we define Iiα as a union of i
3
1
copies of A′i contained in the square 2i+1
, 2i−1
× [0, 1], in such a
way that the first copy is denoted by Ai and the other copies are
denoted by Ak,i with k ∈ {1, 2, · · · , i − 1}. The arc Jiα is the union
of the arcs Ei and Iiα , as in the Figure 2. So Jiα is the arc with an
end point in Ai and the other end point is pα .
156
C. ISLAS
3
1
The sequence of subarcs defined in 2i+1
, 2i−1
× [0, 1] converges
to the arc between aα and cα , and the subarcs Ei converge to the
arc between aα and pα . Let Iα be the arc between aα and cα , and
Iα′ be the arc between pα and aα .
∞
[
We define the fan Fα′ = Iα ∪Iα′ ∪ Jiα . Fα′ contains the harmonic
i=1
fan Fα′′ = Iα′ ∪
∞
[
Ei . The fan that we need is the quotient space
i=1
Fα =
Fα′ /Fα′′ ,
which is, finally, an identification of
Iα′
∪
∞
[
Ei with
i=1
the point aα , and we obtain this fan Fα with aα as its vertex. We
∞
[
will say that Fα is the union Iα ∪
Iiα and Iα ∩ Iiα = {aα }.
i=1
We will denote some important arcs in every Iiα .
For i ∈ N, let Kiα and Lαi denote the subarcs of Ai , [ai , bi ],
and [bi , ai+1 ] , respectively, in Iiα . (We are using the notation of
A′α as the arc between ai to ai+1 to denote these subarcs of Ai .)
Let uαi denote the arc in Iiα joining ai with the first element of
Kiα ∩ Mα . Similarly, let wiα denote the arc in Iiα joining bi with the
first element of Lαi ∩ mα .
α , Lα , uα , and
In the same way, we could define the subarcs Kj,i
j,i
j,i
α
wj,i in Aj,i .
The following lemma is easy to verify and gives important information about Fα .
Lemma 3.1. For Fα , the following hold.
(1) If t ∈ (aα , cα ) and t′ ∈ Iα , then there exist sequences {ti }
and {t′i }, ti , t′i ∈ Iiα , converging to t and t′ , respectively,
such that {d [ti , t′i ]} is bounded away from infinity where d
denotes the length of the arc.
(2) Let {ti } and {t′i } be two sequences, ti , t′i ∈ Iiα , converging
to aα and cα , respectively, then lim {d [ti , t′i ]} = ∞.
(3) For each i ∈ N, let pi ∈ Lαi ; then lim {d [pi , pi+1 ]} = ∞.
(4) Let {pi } and {qi } be two sequences, ti , t′i ∈ Iiα , both converging to aα . If {d [pi , qi ]} is bounded away from zero, then
[aα , bα ] ⊂ lim [pi , qi ].
INCOMPARABLE PLANAR FANS
157
(5) Let {Ai } be a sequence of arcs Ai ⊂ Iiα such that Ai ∩wiα 6= ∅
eventually. If lim Ai ⊃ [aα , bα ], then lim (d (Ai )) = ∞.
(6) Let p ∈ uαki and q ∈ wkαi . Then [p, q] ∩ mα ∩ y = 23 = ∅
(which means that the set of minima in the arc [p, q] does
not include 23 ) iff αki = 0.
Now, we will prove two lemmas which we will use. The proof is
similar to the one appearing in the Awartani’s paper.
Lemma 3.2. Let h : Fα → Fβ be a surjective map and let A be an
arc in Iiα . If B is an arc in h (A), then there exists an arc in A
whose image is B.
Proof: The result follows directly since the map h |A : A → h (A)
is simply a map between closed intervals.
Lemma 3.3. Let h : Fα → Fβ be a surjective map and let {Ai } be a
∞
[
sequence of arcs in
Iiα such that lim sup Ai ⊆ Iα . The following
i=1
hold.
(1) If {d (h (Ai ))} is bounded away from zero, then {d (Ai )} is
bounded away from zero.
(2) If {d (Ai )} is bounded away from infinity, then {d (h (Ai ))}
is bounded away from infinity.
Proof: (1) We assume without loss of generality that {Ai } is a
convergent sequence. Since {d (h (Ai ))} is bounded away from zero,
each h (Ai ) contains a pair of points pi , qi such that lim pi 6= lim qi .
Hence, Ai contains a pair of points p′i , qi′ such that lim p′i 6= lim qi′ ,
which implies that {d (Ai )} is bounded away from zero.
(2) Suppose that lim {d (h (Ai ))} = ∞. Then for each i ∈ N,
h (Ai ) contains a collection {Bi,j : 1 ≤ j ≤ i} of disjoint closed subarcs such that {d (Bi,j ) : (i, j) ∈ N× {1, 2, . . . , i}} is bounded away
from zero. For each i ∈ N and each j, 1 ≤ j ≤ i, let Ai,j be a
subarc of Ai such that h (Ai,j ) = Bi,j . This is possible by Lemma
3.2. It follows from (1) that {d (Ai,j ) : (i, j) ∈ N× {1, 2, . . . , i}} is
bounded away from zero, implying that lim d (Ai ) = ∞.
Lemma 3.4. Let h : Fα → Fβ be a surjective map. Then the
following holds.
(1) h−1 {aβ , cβ } = {aα , cα }.
158
C. ISLAS
(2) If h (aα ) = aβ , then h [aα , bα ] = [aβ , bβ ] and h [bα , cα ] =
[bβ , cβ ].
(3) If h (aα ) = cβ , then h [aα , bα ] = [cβ , bβ ] and h [bα , cα ] =
[bβ , aβ ].
(4) h (bα ) = bβ .
Proof: (1) It suffices to prove that (aα , cα ) ∩ h−1 (cβ ) = ∅ =
(aα , cα ) ∩ h−1 (aβ ). Suppose that (aα , cα ) ∩ h−1 (aβ ) 6= ∅ and
let t′ ∈ [aα , cα ], t ∈ (aα , cα ), be chosen so that h (t′ ) = cβ and
h (t) = aβ . By Lemma 3.1(1), there are sequences {ti } and {t′i } in
∞
[
Iiα , converging to t and t′ , respectively, such that {d [ti , t′i ]} is
i=1
bounded away from infinity. Since lim h (ti ) = aβ and lim h (t′i ) =
cβ , it follows from Lemma 3.1(2) that lim d [h (ti ) , h (t′i )] = ∞.
This contradicts Lemma 3.3(1). Similarly, it can be shown that
(aα , cα ) ∩ h−1 (cβ ) = ∅.
(2) We only prove that h [bα , cα ] = [bβ , cβ ] since the proof that
h [aα , bα ] = [aβ , bβ ] is similar and thus omitted. Since h (aα ) =
aβ , (1) implies that h−1 (cβ ) = cα . Suppose that there exists
t ∈ (bα , cα ) such that h (t) < bβ . Let T denote the sequence
∞
[
Iiα ∩ (y = t) (it means its second coordinate is t). There exi=1
ist two subsequences {pi } and {qi } of T such that for each i ∈ N, pi
and qi are adjacent in T and [pi , qi ]∩Mα = (y = 1), then, the second
coordinate of each one of h (pi ) and h (qi ) is eventually in (aβ , bβ ).
If for an infinity of numbers i, h [pi , qi ] ∩ (y = k) = ∅, k > 12 , we obtain the following. Since cα ∈ lim [pi , qi ], h (cα ) ∈ lim [h (pi ) , h (qi )]
and we obtain that h (cα ) ∈ [aβ , bβ ], contradicting our assumption
that h−1 (cβ ) = cα . If h [pi , qi ] ∩ (y = k) 6= ∅, k > 12 , we obtain
the following. h (pi ) ∈ Lβki and h (qi ) ∈ Lβri with ri 6= ki . It follows
from Lemma 3.1(3) that lim d [h (pi ) , h (qi )] = ∞ and consequently,
lim (h [pi , qi ]) = ∞. This contradicts Lemma 3.3(2) since {d [pi , qi ]}
is bounded away from infinity.
(3) This proof is similar to the proof of (2) and thus omitted.
(4) It follows directly from (2) and (3).
Lemma 3.5. Let h : Fα → Fβ be a surjective map. If neither α
nor β is eventually constant, then the following hold.
INCOMPARABLE PLANAR FANS
159
(1) h (aα ) = aβ ; h (cα ) = cβ ; h [aα , bα ] = [aβ , bβ ] and h [bα , cα ] =
[bβ , cβ ].
∞
[
Iiα such that lim Ai =
(2) If {Ai } is a sequence of arcs in
i=1
2
[bα , cα ] and A
∩
m
∩
y
=
i
α
3 = ∅, ∀i ∈ N, then h (Ai ) ∩
mβ ∩ y = 23 = ∅ eventually.
Proof: (1) By the previous lemma, it suffices to prove that
h (aα ) = aβ . Suppose that h (aα ) = cβ . It follows from the con∞
[
struction that there exist sequences {pi } and {qi } in
Iiβ such
i=1
that lim pi = lim qi = cβ , lim [pi , qi ] = 23 , cβ , and {d [pi , qi ]} is
bounded away from zero. By Lemma 3.2, we may find a sequence
∞
[
{Bi } of arcs in
Iiα such that h (Bi ) = [pi , qi ]. Since {d [pi , qi ]}
i=1
is bounded away from zero, it follows from Lemma 3.3(1) that
{d (Bi )} is bounded away from zero. Choose p′i and qi′ in Bi such
that h (p′i ) = pi and h (qi′ ) = qi . Since h−1 (cβ ) = aα , it follows that
lim p′i = lim qi′ = aα . We assume without loss of generality that
{Bi } is a convergent sequence.
by Lemma 3.1(4), [aα, bα ] ⊂
2 Hence,
lim Bi . Since lim h (Bi ) = 3 , cβ , it follows that h [aα , bα ] ⊃ 23 , cβ ,
contradicting Lemma 3.4(4). Hence, h (cα ) = cβ and h (aα ) = aβ .
(2) The proof is similar and thus omitted.
Lemma 3.6. Let h : Fα → Fβ be a surjective map, and let {pi }
and {qi } be sequences of points in Fα such that for each i ∈ N,
pi ∈ uαi and qi ∈ wiα . If lim pi = lim qi = bα and neither α nor
β is eventually constant, then there exists an integer j0 such that
β
eventually h (pi ) ∈ uβi+j0 and h (qi ) ∈ wi+j
.
0
Proof: Since h (bα ) = bβ , it follows that lim h (pi ) = lim h (qi ) =
bβ .
[∞ β
First, we will show that h (pi ) ∈
uj eventually. Suppose
j=1
not, then we have the following cases.
Case 1. {d [h (pi , Mji )]} converges to zero for some subsequences
{Mji }∞
i=1 of Mβ . Then it can be shown using the uniform continuity
of h that there exists δ > 0, such that h [bα , bα + δ] ⊆ [aβ , bβ ]
contradicting Lemma 3.5(1).
160
C. ISLAS
Case 2. {d [h (pi , mji )]} converges to zero for some subsequences
{mji }∞
i=1 of mβ . Then, using a reasoning similar to the above, it can
be shown that there exists δ > 0, such that h [bα , bα − δ] ⊆ [bβ , cβ ]
contradicting Lemma 3.5(1).
[∞
wjβ (with wjβ in Ai or Ar,i ) eventually. Let
Case 3. h (pi ) ∈
j=1
m∗i = uαi ∩ mα . Since lim [m∗i , pi ] = [aα , bα ], Lemma
implies
[3.5(1)
∞
β
that lim h [m∗i , pi ] = [aβ , bβ ]. Since h [m∗i , pi ] ∩
wj 6= ∅
j=1
eventually, it follows from Lemma 3.1(5) that lim d (h [m∗i , pi ]) = ∞.
This contradicts Lemma 3.2(2) since {d [m∗i , pi ]} is bounded away
from infinity. Hence,
[∞ theβ only remaining possibility is for h (pi ) to
be eventually in
u .
j=1 j
[∞
wjβ is similar and thus omitted.
The proof that h (qi ) ∈
j=1
Now, we will show that if h (pi ) ∈ uβji eventually, then h (qi ) ∈ wjβi
eventually. Suppose not, then h (qi ) ∈ wkβi where ki 6= ji infinitely
often. Then it can be deduced that h [aα , bα ] ∩ (bβ , cβ ) 6= ∅ contradicting Lemma 3.5(1).
We finally prove that if h (pi ) ∈ uβji eventually, then ji+1 = ji + 1.
Suppose that ji+1 6= ji + 1 infinitely often; then it can be deduced
that h [bα , cα ] ∩ (aβ , bβ ) 6= ∅, contradicting Lemma 3.5(1).
Then there exists an integer j0 such that h (pi ) ∈ uβi+j0 and
β
h (qi ) ∈ wi+j
eventually. This completes the proof.
0
Theorem 3.7. Let h : Fα → Fβ be a surjective map. If neither α
nor β is eventually constant, then α dominates β.
Proof: For each i ∈ N, we choose pi ∈ uαi and qi ∈ wiα such
that lim pi = lim qi = bα . By Lemma 3.6, there exists an inteβ
ger j0 such that eventually h (pi ) ∈ uβi+j0 and h (qi ) ∈ wi+j
. Let
0
{αki } be a tail of the sequence of all zeros in α. By Lemma 2.2,
it suffices to prove that βki +j0 = 0 eventually. Since αki = 0, it
follows from Lemma 3.1(6) that [pki , qki ] ∩ mα ∩ y = 23 = ∅. By
Lemma 3.5(2), h [pki , qki ] ∩ mβ ∩ y = 23 = ∅ eventually and hence,
[h (pki ) , h (qki )]∩mβ ∩ y = 23 = ∅ eventually. Since h (pki ) ∈ uβki +j0
and h (qki ) ∈ wkβi +j0 , we conclude by Lemma 3.1(6) that βki +j0 = 0
eventually.
INCOMPARABLE PLANAR FANS
161
Using Lemma 2.4 and Theorem 3.7, we obtain the main result.
Theorem 3.8. There exists an uncountable collection of planar
fans, no member of which maps onto any other.
Acknowledgment. Thanks to Isabel Puga and Alejandro Illanes
for their very useful help in preparation of this paper. Thanks, too,
to Rocio Leonel and my son Carlos Islas-Leonel for their help with
this paper.
References
[1] Marwan M. Awartani, An uncountable collection of mutually incomparable
chainable continua, Proc. Amer. Math. Soc. 118 (1993), no. 1, 239–245.
[2] B. Knaster, Problémes, P 340. Coll. Math. 8 (1961), 278.
[3] Piotr Minc, An uncoutable collection of dendroids mutually incomparable by
continuous function. Preprint. (Available at http://topology.auburn.edu/
pm/cdendr.pdf).
[4] Vı́tĕzslav Kala, Jan Novák, Pavel Pyrih, Marek Sterzik, Martin Tancer, An
uncountable family of incomparable dendroids. Preprint.
Departamento de Matemáticas; Facultad de Ciencias; UNAM;
Circuito C. U. 04510, México, D. F. México
E-mail address: [email protected]