Dendrites with unique hyperspace

Let Y be a metric continuum. Let C n ( Y ) be the hyperspace of nonempty closed subsets of Y with at most n components. In this paper we show that if X is a dendrite with closed set of end points and C 2 ( X ) is homeomorphic to C 2 ( Y ) , for some dendrite Y, then X is homeomorphic to Y. This completes the work by David Herrera-Carrasco and Fernando Macías-Romero who previously proved the corresponding result for each n ≠ 2 .