A fixed-point anomaly in the plane
We define an unusual continuum $M$ with the fixed-point property in the plane $\mathbb R^2$. There is a disk $D$ in $\mathbb R^2$ such that $M \cap D$ is an arc and $M \cup D$ does not have the fixed-point property. This example answers a question of R. H. Bing. The continuum $M$ is a countable union of arcs.