On unicoherence at subcontinua
A continuum \(X\) is said to be unicoherent at a subcontinuum \(Y\subset X\) if for each pair of proper subcontinua \(A\) and \(B\) of \(X\) such that \(A\cup B= X\) the intersection \(A\cap B\cap Y\) is connected [\textit{M. A. Owens}, Topology Appl. 22, 145-155 (1986; Zbl 0589.54043)]. Several mapping properties of this concept are investigated by the reviewer in [ibid. 33, 209-215 (1989; Zbl 0676.54043)], where it is asked if local homeomorphisms preserve unicoherence at subcontinua. The main purpose of the paper under review is to establish an Eilenberg-type characterization of unicoherence at subcontinua and to give a partial positive answer to the above question, for locally connected continua. Unfortunately, the paper contains serious mistakes and some false statements; also some steps in proofs of other results are stated without any sufficient argument. So, corollaries deduced from these results cannot be considered as proved. (1) Lemma 1 on p. 258 is false. (2) The proof of Theorem 8 on p. 259 is invalid. (3) In the equivalence formulated in Theorem 9, p. 259, one implication is false; the proof of the other one is incomplete. (4) In the proof of Theorem 10, p. 260, the false Lemma 1 is used; so the theorem needs another proof. (5) Corollary 11 on p. 261 is based on the (unproved) Theorem 10. In a forthcoming paper ``Remarks on unicoherence at subcontinua'' (to appear in the same journal) by the reviewer, \textit{W. J. Charatonik} and \textit{A. Illanes}, suitable examples are constructed illustrating the falsehood of certain implications in the reviewed paper, and proofs of some results are repaired either by filling up the gaps, or by giving another argument.
scientific article; zbMATH DE number 1102942