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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 (Cj)j in J is a net in CL (X), we define the ``upper Kuratowski limit'' of (Cj) j in J as the set of all x in X such that every nbhd of x meets frequently the sets Cj, and the ``lower Kuratowski limit'' of (Cj)j in J as the set of all x in X such that every nbhd of x meets eventually the sets Cj. If the upper Kuratowski limit and the lower Kuratowski limit coincide, and we call C their common value, then we say that the net (Cj) j in J 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 Moreover, by an unpublished result of Sierpinski, if we assume the Continuum Hypothesis then the Kuratowski convergence on the hyperspace of a metrizable space X is sequentially compact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
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeh-65.