Sequential compactness of the Kuratowski convergence
by
Camillo Costantini
Dipartimento di Matematica dell'Universita` di Torino, Italy

Let CL(X) be the collection of all closed subsets of a topological space X (the ``hyperspace'' of X). If (C<sub>j</sub>)<sub>j in J</sub> is a net in CL (X), we define the ``upper Kuratowski limit'' of (C_j) _{j in J} as the set of all x in X such that every nbhd of x meets frequently the sets C_j, and the ``lower Kuratowski limit'' of (C<sub>j</sub>)<sub>j in J</sub> as the set of all x in X such that every nbhd of x meets eventually the sets C<sub>j</sub>. If the upper Kuratowski limit and the lower Kuratowski limit coincide, and we call C their common value, then we say that the net (C<sub>j</sub>) <sub>j in J</sub> is ``Kuratowski convergent'' to C.

It is an old result of Mrowka that for every topological space X, and every net in CL(X), there is a subnet which is Kuratowski convergent to some C in CL(X) (i.e., the Kuratowski convergence is ``compact''). Another classical theorem in this vein states that if the space X is second countable, then every sequence in CL(X) has a subsequence converging to some C in CL(X) (i.e., the Kuratowski convergence is ``sequentially compact''). Moreover, by an unpublished result of Sierpinski, if we assume the Continuum Hypothesis then the Kuratowski convergence on the hyperspace of a metrizable space X is sequentially compact if and only if X is separable.

We prove that, under Martin's Axiom, the Kuratowski convergence is sequentially compact on the hyperspace of every space X having weight less than the continuum. This shows, in particular, that the result of Sierpinski does not hold in ZFC.

Date received: July 14, 2000