# Hatcher 0.7

Hatcher, Algebraic Topology, Chapter 0

7. Fill in the details in the following construction from [Edwards 1999] of a compact space $Y\subset\mathbb{R}^3$ with the same properties as the space $Y$ in Exercise 6, that is, $Y$ is contractible but does not deformation retract to any point. To begin, let $X$ be the union of an infinite sequence of cones on the Cantor set arranged end-to-end, as in Figure 1 below. Next, form the one-point compactification of $X\times\mathbb{R}$. This embeds in $\mathbb{R}^3$ as a closed disk with curved “fins” attached along circular arcs, and with the one-point compactification of $X$ as a cross-sectional slice. The desired space $Y$ is then obtained from this subspace of $\mathbb{R}^3$  by wrapping one  more cone on the Cantor set around the boundary of the disk.

Figure 1
The spaces mentioned above

Proof.

$\square$