This is an exercise given in order to show that all the hypothesis in the Lefschetz theorem are indeed essential. So I'm given the following three subsets of $\mathbb R ^3$: $$A = \{(x,y,0); x^2 + y^2 = 1\},$$ $$B = \{(x,0,z); (x-1)^2 + z^2 = 1, z \geq 0 \},$$ $$C = \{\left(1+\frac{1}{t}\right)(\cos(2\pi t), \sin(2\pi t), 0); t \in [1, \infty)\},$$
that is, $A$ is a disk in the $xy$ plane, $B$ is a semicircle in the $xz$ plane with center in $(1,0)$ and $C$ is a spiral in the $xy$ plane accumulating on the border of $A$. I've already shown that the space $X = A \cup B \cup C$ is compact and acyclic, now I've been asked to construct a continuous map $f: X \to X$ without fixed points, but every time I think of something it ends up not being continuous. Can anyone think of one such map? I believe this is an exercise from Bredon's Topology and Geometry, by the way.