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Let Y be a metric continuum and let 2X (C( X)) denote the hyperspace of closed subsets (subcontinua) of X. An example is given of a nonlocally connected continuum X such that C( X) is a retract of 2X. By a continuum we mean a compact connected metric space. If X is a continuum, then 2X (respectively, C(X)) denotes the hyperspace of closed subsets (respectively, subcontinua) of X, each with the Hausdorff metric. In this note we will obtain a partial answer to a question raised by Sam B. Nadler, Jr. (8) by giving an example of a nonlocally connected continuum X such that C( X) is a retract of 2X. In 1939 Wojdyslawski (10) proved that C(X) is an absolute retract if and only if X is locally connected. Hence, C(X) is a retract of 2X when Xis locally connected. In Theorem 3.6 of (8), Nadler shows that C(X) is always a continuous image of 2X. Following this result, in Question 3.7 of (8), Nadler asks, "When is C(X) a retract of 2XT Nadler discusses this question further in (6) and in Chapter 6 of (7). In particular, in Theorem 6.11 of (7), Nadler proves a result which indicates that, when X is not locally connected, a retraction from 2X onto C(X), if one exists, must be rather complicated. In (2) the author determined the first partial answer to Nadler's question by giving an example of a nonlocally connected continuum X such that C(X) is not a retract of 2X. In (5) Lawson attempted to give some conditions which would imply the existence of a retraction from 2X onto C(X) for certain nonlocally connected continua. However, as noted in (2), Lawson's proof was not correct. We will show that for one of the examples in Lawson's paper, namely the cone over a convergent sequence, there does exist a retraction from 2X onto C(X). It is interesting to note that this example and the example in (2) are both smooth dendroids. Thus, even within this rather well-behaved class of continua, C(X) may or may not be a retract of 2X when X is not locally connected. We observe in Corollary 1 that the example in this paper also has the property that FX(X), the space of singleton subsets of X, is a retract of 2X. This is the first example of a nonlocally connected continuum for which it has been shown that F,(X) is a retract of 2X. Hence, the example also gives a partial answer to Nadler's question in 6.2 of (7).

A nonlocally connected continuum $X$ such that $C(X)$ is a retract of $2\sp{X}$