On some uniformly pathwise connected continuum
by
Iwona Krzeminska
Technical University of Opole
Coauthors: Janusz R. Prajs

We present an example of a non-g-contractible uniformly pathwise connected continuum P, answering a question of Bellamy in the negative (Houston Problem Book, Problem 64). The same example gives a partial answer to the problem of Nadler concerning the existence of a mapping from the hyperspace C(X) of all nonempty subcontinua of a continuum X, onto X (S. B. Nadler, Jr., Hyperspaces of sets, Marcel Dekker, Inc. 1978, p. 243). Namely, though P is uniformly pathwise connected, it is proved to admit no continuous surjection from C(P) onto P. A natural continuous invariant called \sigma-relative local connectedness has been introduced as a crucial tool for this last result. This invariant has also been employed to obtain a structural characterization of all continua X, whose hyperspace C(X) is the countable union of Peano continua.

Date received: May 14, 1998