Smoothness of hyperspaces and of Cartesian products
by
Wlodzimierz J. Charatonik
University of Wroclaw and Universidad Nacional Autónoma de México

Definition. A continuum X is said to have the property of Kelley if for any x in X, for any continuum K with x in K and any sequence { x_n } tending to x there are continua K_n containing the points x_n and tending to K.

Definition. A continuum X is said to be smooth at a point p in X if for any x in X, for any continuum K with p, x in K and any sequence { x_n } tending to x there are continua K_n containing the points p, x_n and tending to K.

Denote by 2^X the hyperspace of all nonempty compact subsets of X, and by C(X) the hyperpace of all nonempty subcontinua of X.

We will discuss the relations between the notions of the property of Kelley and of smoothness for some continua. In particular we will show that smoothness of the hyperspace 2^X or C(X) implies the property of Kelley for X as well as smoothness of X ×Y implies the property of Kelley for both X and Y.

Date received: March 7, 1997

Copyright © 1997 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 # caal-05.