A Note on the Topology of C-Convergence in Hyperspaces
Proceedings of the American Mathematical Society
In this note we generalize and partially correct a recent Tychonoff theorem for hyperspaces of F. A. Chimenti [lj. For a topological space X, the symbols exp (X), [exp (X)] will denote the hyperspace of all nonempty subsets, of all nonempty closed subsets, respectively, of X. In [l, p. 284], F. A. Chimenti claims the following result: Theorem A. If exp (X.) is equipped with a topology that preserves the C.-convergence for every i £ I, then the product space II. ,exp (X.) z's compact if and only
... compact if and only if the X. are compact. The necessity part of Theorem A is not true, as is seen by choosing the X. noncompact and assigning to each exp (X.) the indiscrete topology. The purpose of this note is to generalize the sufficiency part of Theorem A and to give a corrected version of the necessity part. In [l, p. 283] it is shown that there exist nonindiscrete topologies on exp (X) preserving C-convergence. It is clear that there exists a largest topology, denoted Tc, on exp (X) preserving C-convergence. We will say that a subset A of exp (X) is C-closed if no net in A C-converges to an element of exp (X) -J. It is obvious that the set of all C-closed subsets of exp (X) defines a topology T on exp (X) such that a subset of exp (X) is T -closed if and only if it is C-closed. The lower semifinite topology TL on exp (X) is the topology having as open subbase the subsets of exp (X) of the form [A: A n U 4 0!> where 7/ is open in X [3, p. 179]. It is clear that TL preserves C-convergence, that is, T, C Tc. Of the following four properties, only the last requires a formal proof, in which case, we apply the argument of Theorem 4.2 of [3, p. 161]: (1) T = Tc. In fact, it suffices to note that T preserves C-convergence. (2) If [exp (X)] C J C exp (X), then the topology induced on J by Tc is the largest topology on J preserving C-convergence.