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 $X$, $Y$ 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 $Z$.
