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ABSTRACT. We demonstrate that Proposition 3.1 of [Eric L. McDowell and B. E. Wilder, The connectivity structure of the hyperspaces $C_{\epsilon}(X)$, Topology Proc. 27 (2003), no. 1, 223–232] is false by constructing a locally connected metric continuum which admits a non-locally connected small-point hyperspace.

Let $X$ be a continuum with metric $d$. For any $\epsilon > 0$ the set $C_{d,\epsilon}(X) = {A \in C(X) : \text{diam}d(A) \le \epsilon}$ is called a small-point hyperspace of $X$. The notation $C{\epsilon}(X)$ is used when the metric on $X$ is understood.

Proposition 3.1 of [2] asserts that $X$ is locally connected if and only if $C_{\epsilon}(X)$ is locally connected for every $\epsilon > 0$. While it is true that the local connectivity of $C_{\epsilon}(X)$ for every $\epsilon > 0$ implies the local connectivity of $X$, we show in this note that the reverse implication is false.

Below we construct a locally connected continuum $X$ in $\mathbb{R}^3$ for which $C_{\epsilon}(X)$ fails to be locally connected for some $\epsilon > 0$. The metric considered on $X$ is the usual metric inherited from $\mathbb{R}^3$. All

2000 Mathematics Subject Classification. Primary 54F15; Secondary 54B20. Key words and phrases. cyclic connectedness, hyperspace, locally connected continuum.

The author is grateful to Professor Sam B. Nadler, Jr. for questioning the validity of the proposition that this note addresses. The author is also grateful to the referee for suggestions which significantly enhanced this paper.

points $(r, \theta, z)$ are described using the standard cylindrical coordinate system, and all concepts and notation which are used without definition can be found in [3]. The example is similar to [4, Example 2].

Example 1. For each $n = 1, 2, \dots$, let $S_n$ denote the circle described by ${(1, \theta, n^{-1}) : 0 \le \theta < 2\pi}$ and let $S_0 = {(1, \theta, 0) : 0 \le \theta < 2\pi}$. For each $n = 1, 2, \dots$ and each $i = 1, 2, \dots, 2^n$, let $A_i^n$ denote the straight line segment given by ${(1, 2\pi i/2^n, z) : 0 \le z \le n^{-1}}$. Define $X$ to be the continuum given by

$X = \left( \bigcup_{n=0}^{\infty} S_n \right) \cup \left( \bigcup_{n=1}^{\infty} \bigcup_{i=1}^{2^n} A_i^n \right).$

It is straightforward to show that $X$ is a Peano continuum. We will now prove that $C_\epsilon(X)$ fails to be locally connected at the point $S_0$ when $\epsilon = 2$.

Let ${U_1, \dots, U_k}$ be an open cover of $S_0$ with the property that for every $n = 0, 1, \dots$ and every $i = 1, \dots, k$ it is true that

$(1) \quad S_n - U_i \text{ is connected and has arc length greater than } 3\pi/2.$

Observe that $\mathcal{U} = \langle U_1, \cdots, U_k \rangle$ is an open subset of $C(X)$ that contains $S_0$ as well as all $S_n$ for $n$ sufficiently large. Select $N$ such that $S_N \in \mathcal{U}$. We will prove that $C_\epsilon(X)$ fails to be locally connected at $S_0$ by showing that every arc in $\mathcal{U}$ with endpoints $S_0$ and $S_N$ must contain a point of diameter greater than 2. Let $f: [0, 1] \to \mathcal{U}$ be an embedding for which $f(0) = S_0$ and $f(1) = S_N$. Let $\pi: X \to S_N$ denote the natural projection map. For any subset $S \subset X$ we say that $(1, \theta, z) \in S$ is an antipodal point of $S$ provided that $(1, \theta + \pi, z')$ belongs to $S$ for some $z'$. We will denote the set of antipodal points of $S$ by $\mathrm{AP}(S)$. We now show that

$(2) \quad (1, \theta, z) \in \mathrm{AP}(S) \text{ if and only if } (1, \theta, N^{-1}) \in \mathrm{AP}(\pi(S)).$

To see (2), let $S \subset X$ and let $(1, \theta, z) \in \mathrm{AP}(S)$. By definition it follows that $(1, \theta + \pi, z')$ belongs to $S$ for some $z'$; thus, $\pi(1, \theta + \pi, z') = (1, \theta + \pi, N^{-1})$ belongs to $\pi(S)$. Since $(1, \theta, N^{-1}) = \pi(1, \theta, z) \in \pi(S)$, it follows that $(1, \theta, N^{-1}) \in \mathrm{AP}(\pi(S))$. The argument for the converse is similar.

If $M \in \mathcal{U}$ and $M \subset S_N$, then there exists an arc $A$ (possibly empty) such that $M$ is the closure of $S_N - A$; thus, the only elements ---PAGE_BREAK---

of $M - AP(M)$ are the points that are diametrically opposed to the interior points of A. Therefore, $AP(M)$ is either $S_N$ (if $A = \emptyset$) or the union of two disjoint arcs. Since $f(t)$ is a continuum for each $0 \le t \le 1$, it follows from continuity that

(3) $AP(\pi(f(t)))$ is either $S_N$ or the union of two disjoint arcs.

Continuity also shows that the intersection of $\pi^{-1}(AP(\pi(f(t)))))$ and $f(t)$ is closed; moreover, it follows from (2) that this intersection is equal to $AP(f(t))$. Therefore, we have that

(4) $AP(f(t))$ is closed for every $0 \le t \le 1$.

Suppose that $(1, \theta, z) \in AP(f(t))$; then $(1, \theta + \pi, z') \in f(t)$ for some $z'$. If $z' \neq z$, then $(1, \theta, z)$ and $(1, \theta + \pi, z')$ are more than two units apart. Moreover, if $(1, \theta, z) \in AP(f(t)) - \bigcup_{n=0}^{\infty} S_n$, then it follows from the connectivity of $f(t)$ that there must exist some $z'' \neq z$ with $(1, \theta + \pi, z'') \in f(t)$. It follows that

(5) if $AP(f(t)) - \bigcup_{n=0}^{\infty} S_n \neq \emptyset$ then $\text{diam}(f(t)) > 2$.

We now show that there exists some $t_0 \in [0, 1]$ for which the diameter of $f(t_0)$ is greater than 2. Begin by defining

$t' = \min\{t : [0, 1] : AP(f(t)) \cap S_N \neq \emptyset\}.$

Suppose that $t' = 1$. Choose $\gamma > 0$ small enough such that the $\gamma$-ball, $\mathcal{B}$, about $S_N$ has the properties that $\mathcal{B} \subset \mathcal{U}$ and $S_n \cap (\cup \mathcal{B}) = \emptyset$ for all $n \neq N$. Choose $\delta > 0$ such that if $t \in (1 - \delta, 1]$ then $H_d(f(t), S_N) < \gamma$. Let $t_0 \in (1 - \delta, 1)$. By (3) we have that $AP(f(t_0)) \neq \emptyset$. However, since $t_0 < t'$ we have by the definition of $t'$ and our choice of $\gamma$ that $AP(f(t_0)) - \bigcup_{n=0}^{\infty} S_n \neq \emptyset$. Therefore, $\text{diam}(f(t_0)) > 2$ by (5).

Now suppose that $t' < 1$. Let $q = (1, \theta, z) \in AP(f(t')) \cap S_N$ and let $q' \in f(t') \cap \pi^{-1}(1, \theta+\pi, z)$. We may assume that $q' = (1, \theta+\pi, z)$ since $d(q, q') > 2$ otherwise. Using (3), we have that $AP(\pi(f(t'))) contains an arc $I$ containing $q$. We suppose first that $q$ is an isolated point of $AP(f(t'))$. Let ${y_i}{i=1}^{\infty}$ be a sequence in $I$ converging to $q$; then use (2) to select $x_i \in \pi^{-1}(y_i) \cap AP(f(t'))$ for each $i = 1, 2, \dots$. We have by (4) that $AP(f(t'))$ is closed; hence, some subsequence of ${x_i}{i=1}^{\infty}$ converges to a point $x_0$ of $AP(f(t'))$. Moreover, since ${y_i}_{i=1}^{\infty}$ converges to $q$, we have that $x_0 \in \pi^{-1}(q)$. Finally, since $q ---PAGE_BREAK---

is an isolated point of $AP(f(t'))$, it follows that $x_0$ is a member of $f(t') \cup \pi^{-1}(q)$ that does not belong to $S_N$. Therefore, $d(x_0, q') > 2$, and thus, $\text{diam}(f(t')) > 2$. On the other hand, if $q$ is not an isolated point of $AP(f(t'))$, then we may assume that the arc $I$ containing $q$ belongs to $S_N \cap AP(f(t'))$. Choose $\gamma > 0$ small enough so that (i) no $\gamma$-ball about a point of $I$ meets any $S_n$ for $n \neq N$ and (ii) the midpoint $m = (1, \mu, z)$ of $I$ is not contained in the $\gamma$-balls about the endpoints of $I$. Choose $\delta > 0$ such that if $t \in (t' - \delta, t']$, then $H_d((f(t), f(t')) < \gamma$. Let $t_0 \in (t' - \delta, t')$. Since $H_d(f(t_0), f(t')) < \gamma$, we have by (i), (ii), and the construction of $X$ that $f(t_0)$ contains a point $m'$ for which $\pi(m') = m$; furthermore, we have by (i) that $m' \in S_N$. Thus, $m' = (1, \mu, z) = m \in f(t_0)$. By a similar argument we can show that $(1, \mu + \pi, z) \in f(t_0)$. Therefore, $m \in AP(t_0)$, contrary to our assumption that $t_0 < t'$.

Example 2. K. Kuratowski [1, p. 268] describes a continuum, K, consisting of the segment {(x, 0) : 0 ≤ x ≤ 1}, of the vertical segments {(m/2n+1, y) : 0 ≤ m ≤ 2n+1, 0 ≤ y ≤ 1/2n} and of the level segments {(x, 1/2n) : 0 ≤ x ≤ 1}, where n = 1, 2, .... We note that K is similar in structure to the continuum in the previous example; however, C_ρ₁,ε(K) is locally connected when ρ₁ is the usual metric inherited from R². (Informally, observe that if a subcontinuum A of K is contained in an open subset U of C(X), then U also con- tains subsets of A with diameter smaller than that of A. By first shrinking A to a continuum with smaller diameter within U, one can then continuously grow continua to include a subset of a target subcontinuum within U before continuously releasing A.)

Instead of considering the usual metric on $K$, let $h: K \to S^1 \times [0, 1]$ be an embedding which sends the leftmost vertical segment of $K$ to ${(1, 0, z) : 0 \le z \le 1}$ and the rightmost vertical segment of $K$ to ${[1, 3\pi/2, z) : 0 \le z \le 1}$, and which preserves the vertical and horizontal orientations of all subsets of $K$. Let $d$ denote the usual metric for $h(K)$ inherited from $\mathbb{R}^3$, and let $\rho_2$ denote the metric on $K$ given by $\rho_2(x, y) = d(h(x), h(y))$. Then an argument essentially identical to the one given in Example 1 can be used to show that $C_{\rho_2, \epsilon}(X)$ fails to be locally connected for $\epsilon = 2$.

Noting that the small-point hyperspaces of the arc, circle, and simple triod are all locally connected, while the examples provided ---PAGE_BREAK---

in this article admit non-locally connected small-point hyperspaces, the referee suggests the following question.

Question 1. Are the small-point hyperspaces of an hereditarily locally connected continuum always locally connected?

Recall that a continuum is said to be cyclicly connected provided that any two points of the continuum are contained in some simple closed curve. Theorem 3.11 of [2] states that $C_{\epsilon}(X)$ is cyclicly connected for every $\epsilon > 0$ whenever $X$ is locally connected; however, the argument that is used to justify this assertion uses Proposition 3.1 of [2]. Therefore, the following question remains open.

Question 2. If $X$ is a locally connected continuum with metric $\rho$, must $C_{\rho,\epsilon}(X)$ be cyclicly connected for every $\epsilon > 0$?

REFERENCES

[1] K. Kuratowski, Topology. Vol. II. New edition, revised and augmented. Translated from the French by A. Kirkor. New York-London: Academic Press and Warsaw: PWN, 1968.

[2] Eric L. McDowell and B. E. Wilder, The connectivity structure of the hyperspaces C_ε(X), Topology Proc. 27 (2003), no. 1, 223-232.

[3] Sam B. Nadler, Jr. Continuum Theory: An Introduction. Monographs and Textbooks in Pure and Applied Mathematics, 158. New York: Marcel Dekker, Inc., 1992.

[4] Sam B. Nadler, Jr. and Thelma West, Size levels for arcs, Fund. Math. 141 (1992), no. 3, 243–255.

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE; BERRY COL- LEGE; MOUNT BERRY, GEORGIA 30149-5014

E-mail address: emcdowell@berry.edu