OPEN RETRACTIONS OF CONTINUA

The origin of this researchl was the following problem raised by A. R. Stralka in 1972: Is it true that the only open monotone retraction of a dendroid onto an arc is the identity mapping? (Compare [1], p. 266.) A dendroid is an arcwise connected hereditarily unicoherent continuum. By a continuum we mean a connected compact metric space, and a mapping is meant to be a continuous function. Each dendroid is a hereditarily decomposable continuum. A non-trivial open monotone mapping of an irreducible hereditarily unico herent continuum onto an arc has been constructed by Knaster [3], and Dyer [2] has shown that each such a mapping must possess point-inverses which are indecomposable continua. Although the solution of Stralka's problem is in the negative (see 2.1 below), we offer a number of positive results under some additional conditions imposed upon continua or mappings (see 1.1-1.6). A complete version of the present paper wlll be pub lished elsewhere. The author wishes to thank Professor A. Lelek for continuous encouragement and valuable improvements. 1. Theorems We use a method developed by Dyer [2] to prove the fol lowing result. 1.1. Theorem. Let r: X + Y be an open retraction of