ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 36, Number 3, 2006 LOCAL CONNECTEDNESS IN HYPERSPACES JANUSZ J. CHARATONIK AND ALEJANDRO ILLANES ABSTRACT. Variants of local connectedness as local con- nectedness, local arcwise connectedness, strong local connect- edness and strong local arcwise connectedness at a point are studied for the following hyperspaces of a compact Hausdorff space X: C n (X), C ∞ (X), F n (X), F ∞ (X) and 2 X . 1. Introduction. In [20] it is proved that if X is a Hausdorff com- pact space, then local connectedness and local arcwise connectedness of the hyperspace C (X) of all subcontinua of X and of the hyperspace 2 X of all nonempty closed subsets of X are equivalent at any point. In [21] it is shown that for metric spaces the above properties are equivalent to another one, namely to local k-connectedness. In the present pa- per the above equivalences are studied for further hyperspaces: C n (X) of all members of 2 X that have no more than n components, C ∞ (X) of all members of 2 X that have finitely many components, F n (X) of all members of 2 X that have no more than n points, and F ∞ (X) of all members of 2 X that consist of finitely many points. The obtained results complete not only the above mentioned papers [20] and [21], but also a number of other ones related to the same topic of local con- nectivity properties of hyperspaces as, e.g., [4 7, 9 12, 18, 19] and others. The paper consists of six sections. In the first one we collect, for reader information and completeness of this paper, some known results about local connectedness at a point of the hyperspace 2 X , i.e., at a nonempty closed subset of X. The second and the third sections are devoted to variants of local connectedness at a point of the hyperspace C n (X). We study local connectedness, local arcwise connectedness, strong local connectedness and strong local arcwise connectedness of We regret to inform you that Prof. Janusz J. Charatonik died in July 2004. 2000 AMS Mathematics Subject Classification. Primary 54B20, 54D05, 54F35. Key words and phrases. Connected, continuum, hyperspace, local arcwise connectedness, local connectedness, local n-connectedness, strong local arcwise connectedness. Received by the editors on September 19, 2002, and in revised form on April 20, 2004. Copyright c 2006 Rocky Mountain Mathematics Consortium 811