Borsuk's quasi-equivalence is not transitive
Volume 197 / 2007
Fundamenta Mathematicae 197 (2007), 215-227
MSC: Primary 54C99; Secondary 55P55.
DOI: 10.4064/fm197-0-9
Abstract
Borsuk's quasi-equivalence relation on the class of all compacta is considered. The open problem concerning transitivity of this relation is solved in the negative. Namely, three continua , Y and Z lying in \mathbb{R}^{3} are constructed such that X is quasi-equivalent to Y and Y is quasi-equivalent to Z, while X is not quasi-equivalent to Z.