Hatcher, Algebraic Topology, Chapter 0
7. Fill in the details in the following construction from [Edwards 1999] of a compact space with the same properties as the space in Exercise 6, that is, is contractible but does not deformation retract to any point. To begin, let 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 . This embeds in as a closed disk with curved “fins” attached along circular arcs, and with the one-point compactification of as a cross-sectional slice. The desired space is then obtained from this subspace of by wrapping one more cone on the Cantor set around the boundary of the disk.