#### Glasnik Matematicki, Vol. 37, No.2 (2002), 347-363.

### THE HYPERSPACE C_{2}(*X*) FOR A
FINITE GRAPH *X* IS UNIQUE

### Alejandro Illanes

Instituto de Matematicas, Circuito Exterior,
Cd. Universitaria, Mexico, 04510, Mexico

*e-mail:* `illanes@matem.unam.mx`

**Abstract.** Let *X* be a metric continuum. Let
C_{2}(*X*) be the
hyperspace of *X* consisting of all the nonempty and with at most
two components closed subsets of *X*, with the Hausdorff metric.
In this paper we prove that if *X* is a finite graph and
*Y* is a
metric continuum such that C_{2}(*X*)
is homeomorphic to C_{2}(*Y*),
then *X* is homeomorphic to *Y*.

**2000 Mathematics Subject Classification.**
54B20.

**Key words and phrases.** Continuum, finite graph,
hyperspaces, unique hyperspace.

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