Advances in Pure Mathematics, 2012, 2, 133-138
http://dx.doi.org/10.4236/apm.2012.22020 Published Online March 2012 (http://www.SciRP.org/journal/apm)
Making Holes in the Hyperspace of Subcontinua of
Some Continua
José G. Anaya, Enrique Castañeda-Alvarado, Fernando Orozco-Zitli
Facultad de Ciencias, Universidad Autónoma del Estado de México, Toluca, México
Email: {jgao, eca}@uaemex.mx, forozcozitli@gmail.com
Received November 25, 2011; revised December 23, 2011; accepted December 30, 2011
ABSTRACT

A
CX, Let
X
be a metric continuum. Let
A
is said to make a hole in
CX, if is not unico-
herent. In this paper, we characterize elements

ACX
A
CX such that
A
makes a hole in , where

CX
X
is either
a smooth fan or an Elsa continuum.
Keywords: Continuum; Elsa Continuum; Fan; Hyperspace; Property b); Unicoherence; Whitney Map
1. Introduction
A connected topological space Z is unicoherent if when-
ever =
Z
AB
, where
A
and are connected and
closed subsets of
B
Z
, the set
A
B is connected. Let
Z
be a unicoherent topological space and let be an
element of
z
Z
. We say that makes a hole in z
Z
if

z is not unicoherent. A compactum is a nonde-
generate compact metric space. A continuum is a con-
nected compactum with metric . Given a continuum
d
X
, the hyperspace of all nonempty subcontinua of
X
is denoted by and it is considered with the Haus-
dorff metric. It is known that the hyperspace
CX
CX is
unicoherent (see [1, Theorem 19.8, p. 159]).
In the papers [2] and [3] the author present some par-
tial solution to the following problem.
Problem. Let
X
be a hyperspace of
X
such
that
X

is unicoherent. For which elements,
A
X, does
A
make a hole in
X
?
In the current paper we present the solution to that pro-
blem when
X
is either a smooth fan or an Elsa con-
tinuum and
 
=
X
CX
.
2. Preliminary
We use and to denote the set of positive integers
and the set of real numbers, respectively. Let
Z
be a
topological space and let
A
be a subset of
Z
. We
denote
intA the interior of
A
in
Z
. An arc is any
homeomorphic space to the closed unit interval .
Let in a topological space
[0,1]
,pq
Z
,
,pq will denote
an arc, where and are the end points of
p q
,pq .
A free arc in a continuum
X
is an arc
,pq such that
pq p
,q, is open in
X
. A point in a connected
topological space
Z
is a cut point of (non-cut point of)
Z provided that
z
1
:
is disconnected (is connected).
A map is a continuous function. A map
f
ZS,
where
Z
is a connected topological space and is
the unit circle in the Euclidean plane , has a lifting if
there exists a map such that
1
S
2
:hZ=exp
f
h
exp 1
S
,
where is the map from onto defined by
exp=cos 2π,sin2πttt. A connected topological
space 1
:
Z
has property b) if each map
f
ZS has
a lifting.
By an end point of
X
, we mean an end point in the
classical sense, which means a point of
p
X
that is a
non-cut point of any arc in
X
that contains . A sub-
space of a topological space
p
Y
Z
is a deformation
retract of
Z
if there exists a map
H
:ZI Z such
that, for each
z
x
Z
,
,0 =xx
,

1=
H
H
ZY and,
for each
,1 =
y
Y
H
, yy zZ
. We say that a topolo-
gical space Z is contractible if there exists
, such
that
z is a deformation retract of
Z
. It is known that
each contractible normal topological space has property
b), and so it is unicoherent (see [4, Theorems 2 and 3, pp.
69 and 70]).
3. Smooth Fans
A point of a continuum p
X
is a ramification point
provided that is a point which is a common end point
of three or more arcs in
p
X
that are otherwise disjoint.
A fan is an arcwise connected, hereditarily unicoherent
continuum with exactly one ramification point (here-
ditarily unicoherent means each subcontinuum is uni-
coherent). The ramification point of a fan will be called
the vertex of the fan. If
X
is a fan and
x
yX
, then
,
C
opyright © 2012 SciRes. APM
J. G. ANAYA ET AL.
134
Copyright © 2012 SciRes. APM
,
x
y denotes the unique arc joining
x
and . A fan y
X
with vertex v is said to be smooth provided that if
is a sequence in

nn
x
=1
X
such that it converges to a
point
x
X


=1
,nn
vx
, then the sequence converges
to
,vx in CX.

To establish some notation, let
X
be a smooth fan
with vertex and let i be its end-
points set, where is an infinity indexing set. It fol-
lows from definition of smoothness that the set:
v

EX


=:ie

x X=,Xv:xNC

is a natural homeomorphic copy of
X
in
CX. By
the smoothness of
X
, we have that the set:

=,C v
i
e
i
X

CX
TC


is a closed subspace of . Furthermore, each hy-
perspace
,Cvi
e is a 2-cell and


,=e



j
v,v
i
C
v
Cve for each which ,ij

CX


,CvX
are different. The set of all elements of such that
it contains will be denoted by .
Let
A
CX. We say that
A
is a simple arc if
A
is an arc such that and, there exists a
sequence of satisfying the following
properties:

X
=AE
CX

n
A
im
=1n
1) =l n
A
A
n
n
A

n
A
AA
and
2) for each ,
v
int 
a) ,
b) and
c) .
n
Since
X
is embeded in the Cantor fan (see [5]), we
can regard
X
as embedded in the Euclidean plane
such that v and each i is a convex arc,
where . Note that for , i for
each i. Throughout this section will de-
note the map from
2
]
i=re v
eEXh
=0
eE
,0
X
[,ve
=0r
0,1X onto TC defined
by

X

,=tx,hx
t x. We assume in this section that if
,,
i
eabv , then the distance between and v a
bis less than the distance between and .
v
Lemma 3.1. Let
X
be a smooth fan with vertex .
If v
,ab
i
eE
is an arc contained in , where
, then: 0
i
X,ve

01) If

,b

=1
nn
x
=int a, there exists a sequence
of
X
such that
,=mvb vli ,
n
x and, for each
n, 0
,
ni
x
ve


.
2) If
0
i and
,ae
b
,binta, then 0
,i
be
is a free arc in
X
.
Proof. The proof of (1) is easy.
In order to prove (2), we suppose that is not
a free arc in
0
,i
be


X
. Then there exists
00
,,
ii
e be



0
yb
such that
00
,i
yintbe
. Hence,
00
,i
yintve
=1
nn
y
0
,i
.
Then, there exists a sequence

of
X
ve
0=lim n
yy
such that . Since
X
a smooth fan,
0
,=lim,
n
vy vy
. Notice that
0
,,,ab vbvy .
Let
,zintab

=1
nn
z
0. There exists a sequence
of
X such that 0 and, for each n, =lim n
zz
,
n
zvy
n
. Clearly

,vintab0
zv. Hence,
. Let
>0
be such that
vBz
0 and
0,Bz ab
0
n. Let


0
00
,,
ni
zBz abve
be large enough such that

00 0
,,
ni n
zve vy
. Thus,
 

 
. Since
X
is a fan and
00
,
ni
yXve

0=
n
zv
Q


1,1, 1,Q
qQ
, , this is a contradiction.
Since the Hilbert cube, , is homogeneous (see [1,
Theorem 11.9.1, p. 93]) and is con-
tractible, we have the following result.
Lemma 3.2. Let
. Then has property
b).

Qq
Theorem 3.3. Let
X
be a smooth fan with vertex .
If v
A
is a subcontinuum of
X
such that vA
and,
for each
eEX
i,
,i
A
ve, then
A
does not
make a hole in
CX


.
Proof. We are going to prove that
,CvX A

CX A
is a deformation retract of . Notice that, for
each
BTCX
, there exists
,0,1
BB
xt X
such that
,=
BB
hx tB
 

 
,if ,,
=, 1,,if .
BBB
BBCvX
HBt ttxx BTCX

. We define


Clearly
H
is a map. Then,


,X ACv is a
deformation retract of

CX A. Since Q is ho-
meomorphic to
,CvX (see [6, Theorem 3.1, p. 282]),
{},CvX A has property b) (see Lemma 3.2).
Therefore
CX A has property b) (see [2, Propo-
sition 9, p. 2001]).
Lemma 3.4. Let
X
be a smooth fan with vertex
v

and let
0
,,
i
abCve
be a simple arc contained in
X
, for some
0
i
eEX. Then

00
,=,
ii
intb eb eb

.
Proof. Since
,ab

A is a simple arc, there exists a
sequence =1
nn
of
CX n
that satisfies the required
properties of the definition. Notice that, for each
,
n
A
TCX
0
,
ni
and
A
ve


. Given n
n
a0
,
ni
bve
, let
,
such that
=,
nnn
A
ab.
We need to prove the following claim.
J. G. ANAYA ET AL. 135
Claim. is a free arc in
0
i
,be

X
.
Let . First, we suppose that there exists n0
n
such that
0
n
b

00
,
nn
b

,vb int a
. Since

0
n
b


00
,
ni
be


0
,i
be


and
00
,
in
ea, is a free arc (see (2) of Lem-
ma 3.1). Hence, is a free arc in
X
.
Now, we assume that, for each ,
n
,
ni
bb

00
,,
ii
e be




,>0y0
n

0
e b

 yb. Let . Notice that
=lim n
bb and db. Then there exists

0
,<
n
b d0
,i
ve

,e




0,
n
int A
,
such that . Since is a con-

,bydb
2
vex arc of , we have that

00
,
ni
be
00
ni
yb
00
,
ni
be
. Since
is a free arc in
X
(see (2) of Lemma 3.1).
Thus, is an open subset of

00
,
ni
be
00
,
ni


be
X
such that

0 0
,,
i i
bebe
00
,,
ni
beb




00
,
i
be
00
ni
ey
,
i


.
Hence is an open subset of
be
X
. This
proves the claim.
By Claim, is a free arc in
0
,i
be


X
. Since

,=,
i
ve va


i

=1
mm
y
i

0
,i
intb e


00
, ,ab be



b

00
,i
ntbe


, . Sup-
pose that ei . Then there exists a se-
quence in
X
such that e and, for
each , . Since
0=lim
im
y
i

,>0
i
dbe
0=lim
im
ey m
m0
,
mi
ybe



,=,
i
ve va


00
, ,ab be



, and
0
, we may assume that, for each
e

,
. Since
0
,
mi
yv
X
is a smooth fan,

lim,
im
vy 0,,zbe
0
ve


,=

00 0
,
ii i
be ve. Let




m
z0=limm
zz
.
There exists a sequence such that,
=1m
and, for each ,
m
,zvy
00
,,
ii
be be

mm
is an open set in
. Since


X
and

00
,,
ii
e be


 0
m

0
i
be


,,=
m
vyv

,
ii
b eb


0
zb
zb
, there exists such that
. Then
00
,,
mi
e


 0
,
i
ve
00
mi
zve



int
, this is a contradiction.
Therefore .

,=be


00
Theorem 3.5. Let
X
be a smooth fan with vertex

v. If

A
CX is a simple arc, then
A
makes a
hole in .


CX
0
=, ,
i
AabCve
Proof. We may assume that ,
where
0
i
eEX
and
00,1t0
=atb



such that .
Let:

0
=(,,
i
CvXTCXintbeA



 






and

0
=, 0,1
i
hbeA




0
,i
Xintbe


.
By Lemma 3.4, is a smooth fan.
Then

0
,i
TCX int beA









is a connected
and closed subset of

CX A

CX A


. So, is a con-
nected and closed subset of .
Notice that is homeomorphic to
0
0
,0,1,
i
be bt





. Since 0
0
,0,1,
i
be bt



is a connected subset of


0
0,1 ,
X
bt

CX A
, we have
that is a connected subset of . Clearly
is a closed subset of

CX A.
,=CX abNotice that






and:


00
=0,1, ,0
i
hbbt hbe









.
Let:


100
0
=0,, ,0
i
hb tbthbe

 



and

200
=,1,hb tbt
12
=
.
Clearly, is a separation of
.
Then
CX Av
is not unicoherent.
Theorem 3.6. Let X be a smooth fan with vertex ,
let
0
i
eEX

00
,ii
avee



0
,i
ae
and let . Then

CX

does not make a hole in .
Proof. Let


00
:,[0,1],
ii
GCX aeCX ae



be defined by:
,= :GAttaa A
G
G
.
It is easy to prove that is well defined. In order to
show that is continuous, we define
:0,1GCX CX


by ,= :GAttaa A
G


=1
,
nn n
At . We
prove that is continuous. Let be a se-
quence in
0,1CX

and

00
,0,1At CX
such
00
,=lim ,
A
that nn
tAt. We suppose that there exists
BCX

=lim ,BGAt
such that nn. We will show
=,BGAt
bB
00 . Let
. Consider two sequences

=1
nn
b
and

=1
nn
a
of
X
such that
Copyright © 2012 SciRes. APM
J. G. ANAYA ET AL.
136
=limb

,bGAt
aX
nn

00
,At
BG

0 0
,GA t
A=1
nn
a
:0,1H
 by
n and, for each , nnn
,
nn
and nn
. Taking subsequences if nece-
ssary, we may assume that there exists 0 such
that . Then . Moreover,
b
=
n
im n
a
=ta b
n
bta
0
aA
aA
0=la
=limta 0
00 and, so bG . This proves
that . Now, let 00 . Then
00
. Then there exists a sequence


00
,At ta
a
in
X
such that and, for each ,
0nn
aA=lim n
aa n
. So
. Since, for each ,
, . Thus .
00
ta
nn
ta =l
i
G
mta

=,At
nn
,
n
A
G
n

tta B
n00 00
Hence, is a map. So
BG
v is a deformation retract
of .


0
,i
ae


0
,i
ae


0
e
CX 
Then is contractible. Therefore
has property b) (see [2, Proposition 9,
CX

,i
CX a
p. 2001]).
Theorem 3.7. Let
X
be a smooth fan with vertex , v
0
i
eE

0
,,
i
C ve


let and let

such that

Xab
0
i,eab and is not a free arc of
0
,i
be


X
. Then
,ab

CX does not make a hole in .
Proof. In light of Proposition 9 of [2, p. 2001], it
suffices to prove that there exist two connected, closed
subsets and
of

,ab
CX which have pro-
perty b) and the intersection of them is connected.
We may assume that there exists
00,1t
0=tb a
0=0t
such that
.
We consider two cases.
Case 1. .
Then
,=,vb
ab . Let
=,Xab

TC

and


,X ab=,Cv
. Clearly has property b).
By Theorem 3.1 of [6, p. 282], is a Hilbert
cube. By Lemma 3.2,


,CvX
has property b). Notice that




,X ab



,=abN C
. Clearly

,Xab


NC is homeomorphic to
X
b

.
Since
X
b is connected,

,ab

is con-
nected. By Proposition 8 of [2],
 

) ,ab



[,]=(CX ab
0>0t has property b).
Case 2. . Consider the following sets:

00
1,bt


=,hX t
and



00
0, ,tbt
=,CvX hX
 .
Clearly and
are connected, closed subsets of


,abCX
 
and

00
,t bt=hX
.
Notice that
is homeomorphic to
X
b

. So,
since
X
b is connected,
is connected.
have
property b). If we define
Now, we are going to prove that and
,, =,1
H
hxt shxtts, we have
1hX


1hX
is a deformation retract of . Since is con-
tractible,
1hX
has property b) (see [2, Proposi-
tion 9, p. 2001]). Hence, has property b) (see [2,
Proposition 9, p. 2001]).
In order to prove that
has property b), note that
,CvX is a deformation retract of
. By Theorem
3.1 of [6, p. 282],
,CvX is homeomorphic to a
Hilbert cube. Thus,
,CvX has property b). Hence,
has property b) (see Proposition 9 of [2, p. 2001]).
Therefore
[,]=CX ab
has property b).
Classification
Theorem 3.8. Let
X
be a smooth fan with vertex
and v
CX. Then
A
makes a hole in
A
CX if
and only if
A
a simple arc.
Proof. Let CX be such that
A
A
makes a hole
in
CX. By Theorem 3 of [2, p. 2001] and by
Theorem 3.3,
A
is an arc ,pq

. By Theorems 3.6 and
00
,, ,
ii
pq veve


 for some 3.7,
0
iX
0
,i
qe
eE , and
X
. In order to prove that
is a free arc in
,pq

=1
nn
a

n
b
is a simple arc, let , be sequen-
=1n
ces in
,,pq pq

0
,i
qe q


=limpa =limb
and , respectively,
such that and q. Then
n n
,= ,
nn
pqlima bn and, for each , ,vabnn
,
,inta b
nn
,,
nn
pqa b. Therefore and
A
is a simple arc.
The sufficiency follows from Theorem 3.5.
4. Elsa Continua
A compactification of
0,
0,
with an arc as the re-
mainder is called an Elsa continuum. The Elsa continua
was defined by S. B. Nadler Jr., in [7]. A particular
example of an Elsa continuum is the familiar sin(1/x)-
continuum. There are uncountably many topologically
different Elsa continua, the different topological types
being a consequence of different ways “patterns
into” the remainder of the compactification [8, p. 184].
Let
X
be a continuum. A Whitney map for
CX is a
continuous function
:[0,1]CX
that satisfies the
following two conditions:
BCX such that
A
,
A
B and 1) for any
<
A
B
A
B

,
,
=0x
for each
x
X and c) 2) =1X
.
A Whitney block in
CX, respectively a Whitney
level in

1,
CX, is a set of the form
s
t
, res-
pectively
1t
01, where
s
t. It is known that
Whitney maps always exist (see [1, Theorem 13.4, p.
107]). Moreover, Whitney blocks and Whitney levels in
Copyright © 2012 SciRes. APM
J. G. ANAYA ET AL. 137
CX
=
are continua (see [1, Theorem 19.9, p. 160]).
Throughout this section
X
IR will denote a
Elsa continuum, where
I
is the remainder of
X
and
is homeomorphic to the half-ray .
R
0,
0,t
 
1
Lemma 4.1. Let be a Whitney map,

:CX
let and let

I
0
A
tCI

. Then


0,tA
1()
=|
CI
1
0
Proof. We consider
has p r operty b).

. It is easy to prove
that

1

10
=0,tA
=0


10
has property b).
Let . Since has property b)
(see [9, 12.66, p. 269]) and

1

11
10


0=,
has property b) (see [2, Proposition 8, p. 2001]).
Let

:0

11
,0
f
tAS
exp=|hf
0:h
be a map. Then there
exists a map such that .
0
Given

10,
A
tCRi
R
, it is an arc contained
in and it is determined by its end point, A, lying
near to the end point of . Let
R
A
be an order arc in
from A to

CX i
A
. Since
has property b),
there exists a map :
A
h
xp= |
A
hf s that euch
0
=hi
and


AA
hi
A
.
We define

A
1
0
,



, .t CR
h

=1n
B
:0ht by
 

0
1
,if ,
,if 0
A
hAA
hA hAA T
In order to prove that is continuous, let
be
a sequence of

0,t0
=
n
limB B
A
1
0 such that
for some


00
0,tA
B
1
B
. We consider two cases.
Case 1. For each , . nn
Since is a closed subset of

0
0,tA
n
B
nn
R
BR
p

=0
n
Bpq
q
([,])
|Cpq
1
 
0
=h B
B
,
. Then .
n
Blimh
Case 2. For each , is an arc contained in
.
We consider two subcases.
Subcase 1. .
0
R
n
Let be such that , where de-
,
notes the end point of . Then
R
is a Whitney
map for

,Cpq
. Since is an arc, it


1
([,])
|0
Cpq



1
, ])
|0,
qt

1[0,]t
=
has property b). By Lemma 4 of [10, p. 254],
([Cp
has property b). Then there exists a
map such that

([ , ])
:|
Cpq
g
([,])
|Cpq
gf

exp and

0
11
=
BB
h igi. Notice that
([ ,Cp

1
] 0q
0
h and

[,
]Cpq
g

1
0
are liftings of



1
[,] 0Cpq

f
and

0
11
=
BB
h igi. Then


 


11
0[,] 0
[,]0=Cpq
Cpq
hg

.
0n. Notice that Given
B
n
h y
B
n
g
are lift-
liftings of
B
n
f
 
and
0
==
Bn
BB BB
nn nn
hihigi
.
Then =
B
n
Bn
hg
. Hence,
 
 
000
0
===lim
=lim=lim .
B
n
Bn n
n
hBh BgBgB
hB hB
BI
,nm
Subcase 2. .
0
We can consider that, for any ,
B
B
nm
ii
, if
nm
.
Since 0n and =limBB
X
is a compact space, we
may assume that there exists 0
0
B such that
0
iB
=lim
B
Bn. We can suppose, taking subsequence if it is
necessary, that there exists a subcontinuum 0
ii
B
of
CX =lim such that 0
B
Bn
. It is easy to show that
0
B
is either an order arc from b to or a one
point-set.
0 0
B
0n, we have Given
B
n
is an homeo-
0, n
B
morphism between and
B
n
. Let
:0,
nn
gB
B
n


be such that

1
=g
Bn
n
.
By Lemma 3.1 of [7, p. 330], we can assume that
X
is a subset of . Let
2




=0
=0 0,
Bn
n
n
DXi B


 



D3
.
Notice that is a subset of the Euclidian space
and
0XD D
is a deformation retract of . Then
has property b).
1
10
:0,
f
We define Dt
  


1
if 0,
,,if ,0,.
n
nBn
xt
fxtgtxt iB



by
It is easy to prove that 1
f
is a map. Since has
property b), there exists a map such that
D
3:hD
3
exp =hff

301
11
1
,0=,0
BB
h fihi. Then and
301
{0} {0}
=
XX
hhf


0
. Thus, given n, it can
prove that




31
0,
0, =B
Bnn BiB
iB nn
n
hhf



. Hence,
=limn
hB hB
0
This proves that
.

10,tA
=
0
Theorem 4.2. Let
has property b).
X
IR be an Elsa continuum
and let
CI. Then
A
does not make a hole in
A
CX
.
Proof. In light of Proposition 8 of [2], it suffices to
prove that there exist two connected and closed subsets
and of
CX A, which have property b)
and the intersection of them is connected.
:0,1CX
be a Whitney map. Let Let
Copyright © 2012 SciRes. APM
J. G. ANAYA ET AL.
Copyright © 2012 SciRes. APM
138

=tA
,


1
=,1tA
and

1
=0,t
A. Clearly

CX A

A



=
,
and are connected and closed subsets of
.

CX
The sufficiency follows from Theorem 1 of [2, p.
2001].
REFERENCES
By Lemma 13 of [2, p. 2004], has property b) and,
by Lemma 4.1, has property b).
In order to show that is connected, notice that
and


1t


A=
1
A
tCI




1
tCR



tCR


t
. By
Corollary 3 of [11, p. 386],
[1] A. Illanes and S. B. Nadler Jr., “Hyperspaces: Fundamen-
tals and Recent Advances,” Marcel Dekker, Inc., New
York, 1999.
[2] J. G. Anaya, “Making Holes in Hyperspaces,” Topology
and Its Applications, Vol. 154, No. 10, 2007, pp. 2000-
2008. doi:10.1016/j.topol.2006.09.017






11
=ttCI



1
and
[3] J. G. Anaya, “Making Holes in the Hyperspace of Sub-
continua of a Peano Continuum,” Topology Proceedings,
Vol. 37, 2011, pp. 1-14.
1
approximates the whole continuum

CI. Hence,

=
is connected.
Theorem 4.3. Let
X
IR be an Elsa continua. If

[4] S. Eilenberg, “Transformations Continues en Circon-
férence et la Topologie du Plan,” Fundamenta Mathe-
maticae, Vol. 26, 1936, pp. 61-112.
A
CX such that
A
is homeomorphic to
X
, then
A
does not make a hole in CX.
Proof. In light of Proposition 2.4 of [3, p. 3], it
suffices to prove that there exists a closed neighborhood
of
[5] C. Eberhart, “A Note on Smooth Fans,” Colloquium Ma-
thematicum, Vol. 20, 1969, pp. 89-90.
A
in such that

CX

A
bd


CX
has property
b) and

CX is connected (bd denotes
the boundary of in ).
CX
[6] C. Eberhart and S. B. Nadler Jr., “Hyperspaces of Cones
and Fans,” Proceedings of the American Mathematical
Society, Vol. 77, No. 22, 1979, pp. 279-288.
doi:10.1090/S0002-9939-1979-0542098-5

:CX0,1
Let


,1I
be a Whitney map. Let
1
=

[7] S. B. Nadler Jr., “Continua Whose Cone and Hyperspace
Are Homeomorphic,” Transactions of the American Ma-
thematical Society, Vol. 230, 1977, pp. 321-345.
doi:10.1090/S0002-9947-1977-0464191-0

. Clearly is a closed neigh-
borhood of
A
. Since



1
=I

CX
bd

,

is connected. By [12, Theorem 4.3, p. 217],
is a 2-cell. Moreover,
CX
bd
A
is an element of its
manifold boundary (see [11, Lemma 2, p. 386]). Then

A
is contractible. Therefore
[8] S. B. Nadler Jr., “Arc Components of Certain Chainable
Continua,” Canadian Mathematical Bulletin, Vol. 14, No.
2, 1971, pp. 183-189. doi:10.4153/CMB-1971-033-8

A
=
has pro-
perty b) (see [2, Proposition 9, p. 2001]). [9] S. B. Nadler Jr., “Continuum Theory: An Introduction,”
Marcel Dekker, Inc., New York, 1992.
Classification [10] A. Illanes, “Multicoherence of Whitney Levels,” Topology
and Its Applications, Vol. 68, No. 3, 1996, pp. 251-265.
doi:10.1016/0166-8641(95)00064-X
Theorem 4.4. Let
X
IR
be an Elsa continuum
and let
A
CX. Then
A
makes a hole in
CX if
and only if
A
is a free arc such that
.
pq

int pq,pq
[11] W. J. Charatonik, “Some Counterexamples Concerning
Whitney Levels,” Bulletin of the Polish Academy of Sci-
ences Mathematics, Vol. 31, 1983, pp. 385-391.
Proof. Let
A
CX be such that
A
makes a hole
in . By Theorem 3 of [2, p. 2001] and Theorems
4.2 and 4.3,

CX
A
is an arc contained in R. So, pq
[12] C. B. Hughes, “Some properties of Whitney continua in
the hyperspace C(X),” Topology Proceedings, Vol. 1, 1976,
pp. 209-219.
A
is a free arc in
X
. By Theorem 4 of [2, p. 2001],
.

q,pqint p