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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 14, no. 1, 2013

pp. 73-84

The hyperspaces Cn(X) for finite ray-graphs

Norah Esty

Abstract

In this paper we consider the hyperspace Cn(X) of non-empty and
closed subsets of a base space X with up to n connected components.
The class of base spaces we consider we call finite ray-graphs, and are
a noncompact variation on finite graphs. We prove two results about
the structure of these hyperspaces under different topologies (Hausdorff
metric topology and Vietoris topology).

2010 MSC: 54B20

Keywords: hyperspaces, finite graphs

1. Introduction

The last thirty years have produced a large amount of research in the area of
hyperspaces. A hyperspace is a topological space whose points are subsets of a
given base space. There are several topologies available for such spaces. If the
base space is compact, two of the most popular topologies, the Hausdorff met-
ric topology and the Vietoris topology, agree. However, when the base space is
not compact, they differ, and in fact the Vietoris topology is non-metrizable.
In contrast, by using a bounded metric on the space, or allowing for infinite-
valued metrics, the Hausdorff topology arises from a metric. Most of the study
of hyperspaces has been done in the case where the base space X is a continuum.

In 1968, Duda did an examination of the hyperspace of subcontinua of fi-
nite connected graphs, and under some minor conditions was able to give a
description of the hyperspace as a polyhedron, decomposable into balls of var-
ious dimensions [1], [2]. A single hyperspace may consist of several sections of
different dimension: a two-dimensional disc glued to a three dimensional ball,



74 N. Esty

etc. In particular, for X a finite graph, the hyperspace of subcontinua is known
to be compact and connected.

In this paper we are interested in the situation where the base space is not
compact. We look at a natural generalization of finite graphs which we call
finite ray-graphs, which consist of vertices, edges, and rays. Because the graphs
are not compact we must always specify which topology we are using, and in
section 5 of this paper we will use first the Hausdorff metric topology, arising
from the Hausdorff metric, which we allow to be infinite-valued; we will call this
simply the Hausdorff topology. In section 6 we will use the Vietoris topology.

To assist the reader, in sections 3 and 4 we present several models of the
hyperspace C(X) of closed, connected subsets of the base space X. We begin
with a few known examples on compact base spaces, in order to provide an
analog for our non-compact examples. In section 4 we state a theorem about
hyperspace C(X ∨p Y ) of a wedge product at a point, when the hyperspaces
of the C(X) and C(Y ) are known. We state this theorem without proof, as it
seems to be well-known in the folk-lore (although we have been unable to find
a reference). This theorem gives a nice algorithm for drawing hyperspaces.

In sections 5 and 6 we prove two main results about the number of connected
components of the hyperspace Cn(X) (closed subsets of X with up to n con-
nected components) of a finite, connected ray-graph X: once in the Hausdorff
topology and once in the Vietoris. In particular, we show that when allowing
for an infinite-valued Hausdorff metric, a finite, connected ray-graph with k
rays will have a hyperspace Cn(X) with 2

k connected components for all n,
and will not be compact. In contrast, under the Vietoris topology Cn(X) is
connected for all n.

2. Preliminaries and Notation

2.1. Notation. There is not always consistent notation used for the different
hyperspaces of a given base space X. We attempt to use those notations from
the literature which are least ambiguous. Given a metric space X, we define
the following notation for the hyperspaces we will discuss:

• CL(X) = {A ⊂ X : A closed and A 6= ∅}
• Cn(X) = {A ∈ CL(X) : A has at most n connected components }
• C(X) = C1(X)
• CA(X) = {B ∈ C(X) : A ⊂ B}.

The final example is called the containment hyperspace. This concept is es-
pecially useful to us when A is a vertex of the graph. If A = {p} we may write
Cp(X) rather than C{p}(X).

It should be pointed out that much of the literature on hyperspaces assumes
that the base space X is compact, in effect making C(X) the hyperspace of



The hyperspaces Cn(X) for finite ray-graphs 75

subcontinua, but we are not assuming that here. This is also why we write
CL(X) rather than 2X, which is more common, but to many readers may
mean bounded closed subsets, which we do not mean. When we wish to refer
to a general hyperspace, we will write H(X).

Initially we will endow our hyperspaces with the Hausdorff topology (τH).
The Hausdorff topology has the virtue that it arises from a metric in our
approach, although since we are interested in unbounded base spaces, we allow
the metric to be infinite-valued. Let H(Y ) be a hyperspace over a metric base
space Y . If A, B ⊂ Y , and if NY (A, ǫ) indicates the ǫ-neighborhood in the
space Y around the subset A, then the Hausdorff distance in the hyperspace is
given by

dH(A, B) = inf{ǫ : A ⊂ NY (B, ǫ) and B ⊂ NY (A, ǫ)}

If the elements of the hyperspace are not closed subsets, then it is possible
to have the distance between two non-equal sets be zero. However we will
deal exclusively with closed sets. One can see from this definition that if A is
bounded and B is not, the Hausdorff distance between A and B is infinite. It is
worth noting that using this infinite-valued metric results in a different picture
of the hyperspaces than one gets from putting a bounded metric on the space
X, which is why we do it.

Later in the paper we will use the Vietoris topology. This topology is usually
given by a basis definition, which we will recall in section 6.

2.2. The class of base spaces: finite ray-graphs. For our base spaces,
we will consider a variation on finite graphs, which we will call finite ray-
graphs. These graphs will consist of a finite number of vertices (points), edges
(homeomorphic to [0, 1] and attached at two vertices, or at one vertex twice)
and rays (homeomorphic to [0, ∞) and attached at one vertex). We will restrict
our attention to finite connected ray-graphs. We give some simple examples of
models for C(X) in sections 3 and 4. The metric on these graphs will be that
of arc-length, and we will consider all edges as having length one. We shall call
the class of all such ray-graphs X and elements of that class X.

3. Some Basic Examples of (C(X), τH)

Here we give a few geometric models of the hyperspace (C(X), τH) for spe-
cific X ∈ X . We will provide a brief explanation but no full proofs for the
examples in sections 3.1, 3.2 and 4.1 as they are well-known. The purpose
of outlining the examples is to use them to compare to their analogous non-
compact counterparts (sections 3.3, 4.2 and 4.3). For more detail or proofs on
the known hyperspaces, see [7].



76 N. Esty

3.1. X ≈ [0, 1]. If X is a segment homeomorphic to [0, 1] then any element
A ∈ C(X) is of the form [a, b], with 0 ≤ a ≤ b ≤ 1. Clearly there is a
homeomorphism from the hyperspace C(X) to the solid triangle in R2 with
vertices at (0, 0), (0, 1) and (1, 1) which takes an interval [a, b] to the point
(a, b). (Here we are abusing the notation to say that [a, a] = {a}.) See Figure 1.
Notice that the left edge of the triangle corresponds to subsets of X which
contain 0, i.e. the containment hyperspace C{0}(X). The top edge corresponds
to subsets which contain 1, C{1}(X), and the hypotenuse corresponds to the
single-element sets. We will refer to this triangle as T .

0 1

X

(0, 0)

(0, 1) (1, 1)

C(X)

A A

Figure 1. X = [0, 1] and C(X), as well as an element A ∈ C(X)

3.2. X ≈ S1. If X ≈ S1, then elements of C(X) can be categorized by their
midpoint and their length. We can make a homeomorphism from C(X) to the
solid unit disc by mapping an arc with length l and midpoint p to the point
which sits on the radial line through p, and whose distance from the origin is
1− l

2π
. See Figure 2. Notice that points on the boundary of the disc correspond

to single-element sets, and the center point of the disc corresponds to the full
circle. We will refer to this disc as D.

Although [0, 1] 6≈ S1, their two hyperspaces D and T are homeomorphic. It
is known that for finite graphs this is the only such example [1].

3.3. X ≈ [0, ∞). Here we give our first non-compact example, as an analogy
to the compact closed interval. If X is a ray homeomorphic to [0, ∞), then
elements of C(X) are either bounded intervals of the form [a, b] or unbounded
intervals of the form [a, ∞). We can make a homeomorphism from C(X) to
the space T ∞ ⊔ [0, ∞), where T ∞ = {(a, b) ∈ R2 : 0 ≤ a ≤ b} is an “infinite tri-
angle.” This is done by mapping [a, b] to (a, b) ∈ T ∞ and [a, ∞) to a ∈ [0, ∞).
See Figure 3.

Notice that although T ∞ is itself unbounded, elements of T ∞ correspond to
bounded subsets of X, and in particular, the left edge of T ∞ corresponds to



The hyperspaces Cn(X) for finite ray-graphs 77

X

A

B

C(X)

A

B

Figure 2. X = S1 and C(X), as well as two elements A, B ∈ C(X)

0

X

(0, 0)

C(X)

0

A

B

Figure 3. X = [0, ∞) and C(X), as well as the elements
A = [.25, .5] and B = [.25, ∞), both in C(X)

bounded elements which contain 0, and the hypotenuse corresponds to single-
element sets. For a fixed horizontal value a, increasing the vertical value b
corresponds to longer bounded intervals. Since the second component, [0, ∞),
corresponds to unbounded intervals, it can be loosely thought of as the “top”
of the infinite triangle. In this example, unlike before, the containment hy-
perspace C{0}(X) has two components: the left edge of the triangle and the
leftmost point 0 ∈ [0, ∞). Clearly this C(X) is not connected and not compact.

With these three examples we can form several more examples by under-
standing what happens to the hyperspace when you form the wedge product
of two graphs.



78 N. Esty

4. The wedge product of graphs

Let X = X1 ∨p X2 be the wedge product of two ray-graphs, where p is
a vertex of both. It is clear that the hyperspace C(X) will contain all the
elements which are in C(X1) and C(X2). It will also contain elements which
correspond to subsets of X that contain the joining point p and part of X1 and
X2. Therefore C(X) will contain a cross product of Cp(X1) and Cp(X2).

Theorem 4.1. For X1, X2 ∈ X and the wedge produce X = X1 ∨p X2 (where
p is a vertex of X1 and X2), then

C(X) ≈
C(X1) ⊔ Cp(X1) × Cp(X2) ⊔ C(X2)

(Cp(X1) ∼ Cp(X1) × {p} and {p} × Cp(X2) ∼ Cp(X2))

This theorem, which seems to be well-known in the folklore (and certainly
applies to a larger class of spaces than graphs), gives us the following nice
algorithm for drawing C(X):

(1) Draw Cp(X1) × Cp(X2).
(2) Attach the rest of C(X1) to the figure by identifying its subset Cp(X1)

with the slice Cp(X1) × {p} in the cross product.
(3) Attach the rest of C(X2) to the figure by identifying its subset Cp(X2)

with the slice {p} × Cp(X2) in the cross product.

We shall use this algorithm in the following examples.

4.1. The noose. The space is the wedge-product of a circle and an interval.
By following the steps outlined above, we get for the hyperspace a solid cylinder
with a disc attached along the bottom and a fin attached along the side. (The
cross-section of the cylinder is in fact a cardiod, but we represent it here as a
circular disc.) See Figure 4.

p

X

p

C(X1)

C(X2)

Cp(X1) × Cp(X2)

C(X)

Figure 4. X a noose, and C(X)



The hyperspaces Cn(X) for finite ray-graphs 79

4.2. The infinite noose. Let X be the infinite noose, made up of X1 ≈ S
1

and X2 ≈ [0, ∞), joined at the point p. Cp(X1), as we have already noted in
section 4.1, is a cardiod inside the unit disc, which we will draw as a subdisc.
Recall from section 3.3 that Cp(X2) has two components: the left edge of the
infinite triangle T ∞, and the left-most point of the ray. The point of C(X2)
which corresponds to the single-point set {p} ⊂ X2 is at the bottom of the
triangle.

Crossing Cp(X1) with Cp(X2), we get an infinite cylinder and a disc. We
attach C(X1) to the slice Cp(X1) × {p}, along the bottom of the cylinder. We
attach C(X2) along {p} × Cp(X2), producing an infinite fin off the side of the
cylinder and a ray off the side of the disc. See Figure 5.

p

X
p

C(X)

Figure 5. X is the infinite noose. C(X) has two components.

4.3. The real line. Let X1 = X2 ≈ [0, ∞), both just a single vertex with a
ray attached. Then we can think of X ≈ R as the result of attaching these two
subgraphs along their vertex. Both subgraphs have hyperspaces which consist
of T ∞ ⊔ [0, ∞), and the containment hyperspace for the vertex is the union of
the left edge of the triangle and the leftmost point of the ray.

Following the algorithm, Cp(X1) × Cp(X2) gives us four components: an
infinite square, two rays, and a point. When we attach the rest of C(X1) along
the correct slice, it attaches the rest of the triangle along one side of the infinite
square, and the rest of a ray along one of the rays. Similarly when we attach
the rest of C(X2) it attaches the rest of an infinite triangle along the other side
of the infinite square, and another ray along the second ray. The end result is
four components: a half-plane, two real lines, and a point.



80 N. Esty

5. The connected components of (Cn(X), τH)

The last two examples of section 4 show that under the Hausdorff topology,
there is a relationship between the number of rays in a given graph, and the
number of connected components of its hyperspace. That relationship is what
we explore in this section.

Let R = {R1, . . . Rk} denote the set of rays in a given ray-graph. If #R = k,
we will call X an k-legged graph. We will denote by XG = X − ∪

k
i=1Ri. If

A ⊂ X, and A ∩ Ri is an unbounded interval, we say that A is unbounded in
direction i. In this way we can talk about the unbounded direction set of A,
which is the set of indices between 1 and k for which A is unbounded in direc-
tion i. Clearly there are 2k possible unbounded direction sets, in one-to-one
correspondence with the power set of {1, 2, . . . , k}.

Let Pk be the power set of {1, 2, . . . , k}. Define a function φ : Cn(X) → Pk
by φ(A) = ∆, where ∆ ∈ Pk is the unbounded direction set of A. Recall that
we denote the Hausdorff distance between two elements A and B by dH(A, B).

Lemma 5.1. Let A, B ∈ Cn(X) under the Hausdorff topology.

(1) If dH(A, B) < ∞, then A and B have the same unbounded direction
set.

(2) If A and B have distinct unbounded direction sets, e.g. there exists a
ray Ri ∈ X such that A is unbounded in direction i but B is not, then
there does not exist any path through Cn(X) from A to B.

Proof. If A is unbounded in direction i and B is not, then clearly dH(A, B) =
∞. Since any path is a continuous image of a compact set, it must have a
compact image which contains A and B. If dH(A, B) = ∞, this is impossible.

�

Lemma 5.2. If X ∈ X is a k-legged graph, then for all n, the hyperspace
(Cn(X), τH) has at least 2

k connected components.

Proof. Let X ∈ X be a graph with k distinct rays, labelled R1, . . . , Rk. Con-
sider the power set Pk of {1, . . . , k}. Each element of the power set corresponds
to an unbounded direction set. We will use the map φ : Cn(X) → Pk from
above, given by φ(A) = ∆ if A is unbounded in the direction set ∆. We will
show that φ is continuous, and therefore Cn(X) has at least 2

k connected com-
ponents.

Because (Cn(X), τH) is first countable, it is enough to show convergent se-
quences are mapped to convergent sequences. Let Am → A be a convergent
sequence of elements of Cn(X), meaning that dH(Am, A) → 0 as m → ∞.
If A is unbounded in direction Ri, and Am is not (or vice versa) we know



The hyperspaces Cn(X) for finite ray-graphs 81

dH(A, Am) = ∞, so for all m greater than some m
∗ we must have Am un-

bounded in the same set of directions as A. Therefore φ(Am) = φ(A) for all
m > m∗ and φ is continuous. �

Theorem 5.3. If X ∈ X is a k-legged graph, then for all n, the hyperspace
(Cn(X), τH) has exactly 2

k path-connected components.

The previous lemma showed that Cn(X) has at least 2
k connected compo-

nents. We will now show that it has no more than that, by showing that for all
∆ ∈ Pk, {A ∈ Cn(X) : φ(A) = ∆} is a path-connected set. This will be done
by taking any element in a given component and constructing a path from it
to a designated “default” element of that component.

A note on notation: a subinterval of A inside the edge Ei will be denoted
[a, b]ei . A subinterval of A inside the ray Ri will be denoted [a, b]

r
i . For rays, the

vertex is 0. For edges which only have one ramification point X, the endpoint
0 is the ramification point.

Proof. Fix n. We begin by choosing for each ∆ ∈ Pk a particular element A∆
of {A ∈ Cn(X) : φ(A) = ∆}. The element A∆ will consist of the complete
finite-graph XG, and all the rays which are in the unbounded direction set ∆,
but no part of the other rays. It will have one connected component. Precisely,

A∆ = XG ∪
⋃

i∈∆

Ri

Given an element A ∈ Cn(X) with φ(A) = ∆, we will construct a path from
A to A∆. There are three steps. First, any sections of A contained completely
in rays Ri where i ∈ ∆ we will grow until they touch XG at the vertex. Sec-
ondly, any sections of A contained in rays Rj where j 6∈ ∆, we will shrink down
until they are gone. Finally, we grow the remaining subset out so that it in-
cludes all of XG. The first two steps of this process will either keep constant or
decrease the number of components of A; the last step will produce an element
with one component. So the path will stay in Cn(X) at all times.

By definition, if i ∈ ∆ then A ∩ Ri 6= ∅. Consider those i ∈ ∆ for which
A ∩ Ri 6= Ri. For each such i, that intersection will be a finite number of
intervals, one of which is unbounded. Call the unbounded one [ai, ∞)

r
i , and

call the vertex where that ray is attached vi. We will grow this interval out so
that it encompasses all of Ri. We define the first step in the path as follows.
f0 : [0, 1] → Cn(X) is given by

f0(t) = A ∪
⋃

i∈∆

[tvi + (1 − t)ai, ∞)
r
i

If A had several intervals contained in that ray, this process will consume
them, reducing the number of components of A. If A had empty unbounded



82 N. Esty

direction set, this will do nothing to A. Let A1 = f0(1).

Now consider those j 6∈ ∆ with A1 ∩ Rj 6= ∅. That intersection will consist
of a finite number of bounded intervals [aij, b

i
j]
r
j (where i = 1, . . . , lj for some

lj ≤ k). We wish to shrink and slide each of those intersections down to the
vertex vj. To do that we define a path f1 : [0, 1] → Cn(X) by

f1(t) = A1 ∪
⋃

j 6∈∆

lj
⋃

i=1

[tvj + (1 − t)a
i
j, tvj + (t − 1)b

i
j]
r
j

This is a path from A1 to the set which agrees with A1 in XG, contains all of
the rays in the unbounded direction set, but does not contain any section of
any rays which are not in the unbounded direction set (apart from possibly the
vertices). Call this second intermediate set A2 = f1(1).

The final step will grow the subset A2 out until it includes all of XG. Fix
an element a ∈ A2 ∩XG. Because XG is a graph, it is path connected, so there
exists a path γ : [0, 1] → XG which starts at a and whose image contains all of
XG. Define

f2(t) = A2 ∪
⋃

x∈[0,t]

γ(x)

Clearly f2(0) = A2 and f2(1) = A∆. The continuity of γ makes f2 continu-
ous, and because components may merge together, but never split apart, the
construction ensures f2(t) ∈ Cn(X) at all times. Following f0 with f1 and f2,
we have a path from A to A∆. Hence the set {A ∈ Cn(X) : φ(A) = ∆} is
path-connected. This completes the proof. �

6. Connectedness of (Cn(X), τV )

In this section we will explore some of the distinctions between a hyperspace
of a finite ray-graph under the Hausdorff topology with that same hyperspace
under the Vietoris topology. We begin by recalling the definition.

6.1. The Vietoris topology. Let X be the base space, and let U1, . . . , Un be
a finite number of open subsets of X. For any hyperspace H(X) over X, we
define the open set U∗ =< U1, . . . , Un > in the following way:

A ∈ U∗ iff

(1) A ⊂
⋃n

i=1 Ui
(2) A ∩ Ui 6= ∅ for all i = 1, . . . , n

Such open sets make a basis for the Vietoris topology on H(X). It is some-
times more useful to treat the topology as the supremum of the upper and
lower Vietoris topologies. The upper Vietoris topology is generated by sets of
the form

U+ = {A ∈ H(X) : A ⊂ U}



The hyperspaces Cn(X) for finite ray-graphs 83

where U is open in X. The lower Vietoris topology is generated by sets of the
form

V − = {A ∈ H(X) : A ∩ V 6= ∅}

where V is open in X. Subbase elements of the Vietoris topology are then of
the form U+ and V −1 ∩ V

−
2 ∩ · · · ∩ V

−
n .

6.2. Path-connectedness. In [4] we proved that CL(M) was contractible for
any Borel compact space M having the property that the closure of open balls
is closed balls. Since ray-graphs satisfy those conditions, we know that CL(X)
is contractible:

Theorem 6.1. (CL(X), τV ) is contractible.

We now prove a companion theorem to Theorem 5.3.

Theorem 6.2. (Cn(X), τV ) is path-connected.

The proof is similar in flavor to the proof of Theorem 5.3. In fact, the first
part is identical: take an element A ∈ C(X) and construct a path from it to the
element A∆, ∆ = φ(A). As it was in the Hausdorff topology, this construction
is continuous in the Vietoris topology. The distinction comes when we then
form a path from A∆ to the element X.

Proof. Recall that if φ(A) = ∆ ∈ Pk, we define the element A∆ as

A∆ = XG ∪
⋃

i∈∆

Ri

Start with the same path f = f2 ◦ f1 ◦ f0 from A to A∆ as given in the proof
of Theorem 5.3. As before, the path remains in Cn(X) at all times, and results
in A∆ ∈ C(X). The proof that this is continuous under the Vietoris topology
is quite similar to the proof that the second path is continuous, so we omit it.

For the second path we will connect A∆ to X. Let f(t) =
t

1−t
, and define a

path γ : [0, 1] → C(X) by:

γ(t) =

{

A∆ ∪
⋃

i6∈∆[0, f(t)]
r
i t ∈ [0, 1)

X t = 1

Obviously γ(0) = A∆ and by construction, γ(t) ∈ C(X) for all t. To show
γ is continuous, it is enough to show that it is continuous with respect to the
upper and lower Vietoris topologies. Note that s < t implies γ(s) ⊂ γ(t). This
means for the upper topology, which is concerned with containment, we only
have to worry about t increasing, and for the lower topology, which is concerned
with intersection, we only have to worry about t decreasing.

We begin by checking continuity with respect to the upper Vietoris topol-
ogy. Fix t0 ∈ [0, 1] and suppose γ(t0) ∈ U

+. If t0 = 1 then γ(t0) = X and
as X ∈ U+ we have U = X, which clearly implies γ(t) ∈ U+ for all t. So we



84 N. Esty

assume that t0 ∈ [0, 1) and that U 6= X. Since U 6= X and γ(t0) ⊂ U, there
exists ǫ = d(γ(t0), U

c) > 0. Continuity of f(t) implies that there exists some
δ, 0 < δ ≤ ǫ such that if |t − t0| < δ then |f(t) − f(t0)| < ǫ. Then the total
growth from γ(t0) to γ(t) is small, i.e. d(γ(t), γ(t0)) < ǫ and hence γ(t) ∈ U

+.
So γ is continuous with respect to the upper Vietoris topology.

Now we check continuity with respect to the lower Vietoris topology. Let
γ(t0) ∈ V

−
1 ∩ V

−
2 ∩ · · · ∩ V

−
n , meaning that γ(t0) ∩ Vi 6= ∅ for each i. Pick

a point xi ∈ γ(t0) ∩ Vi. If xi ∈ A∆ then xi ∈ γ(t) for all t. If not, then
xi ∈ [0, f(t0)]

r
ℓ for some ℓ. Possibly xi is in the interior of that interval, or

possibly it is the endpoint f(t0). If xi is in the interior, i.e. xi < f(t0), then
pick δi > 0 such that |t − t0| < δi implies |f(t0) − f(t)| < f(t0) − xi, and then
xi ∈ f(t) also. If xi = f(t0), then let di = d(xi, V

c
i ) > 0 and choose δi such

that |t − t0| < δi implies |f(t) − f(t0)| < di. Then [0, f(t)]
r
ℓ
∩ Vi 6= ∅. All

together, let δ = min{δi : i = 1, . . . , n}. Then for each i, γ(t) ∩ Vi 6= ∅. So γ is
continuous with respect to the lower Vietoris topology.

Combining with the path f, we have constructed a path from any A ∈ Cn(X)
to X. �

References

[1] R. Duda, On the hyperspace of subcontinua of a finite graph I, Fund. Math. 62 (1968),
265–286.

[2] R. Duda, On the hyperspace of subcontinua of a finite graph II, Fund. Math. 63 (1968),
225–255.

[3] C. Eberhart and S. Nadler, Hyperspaces of cones and fans, Proc. Amer. Math. Soc. 77
(1979), no. 2, 279–288.

[4] N. Esty, On the contractibility of certain hyperspaces, Top. Proc. 32 (2008), 291–300.
[5] A. Illanes, The hyperspace C2(X) for a finte graph is unique, Glasnik Mat. 37 (2002),

347–363.
[6] A. Illanes, Finite graphs X have unique hyperspaces Cn(X), Top. Proc. 27 (2003),

179–188.
[7] A. Illanes and S. Nadler, Hyperspaces: Fundamentals and Recent Advances, Marcel

Dekker, Inc., New York, 1999.

(Received April 2012 – Accepted November 2012)

Norah Esty (nesty@stonehill.edu)
Department of Mathematics, Stonehill College, Easton, Massachusetts 02357


	The hyperspaces Cn(X) for finite ray-graphs. By N. Esty