The slinky

The model below shows a one-dimensional compact metrizable space that has strong computable type, and properly contains a copy of itself. The result is explained in this note.
Each section is an infinite spiral whose two ends accumulate along two red circles. The space is made of two infinite chains of such sections (a blue one and a green one) accumulating along a central orange circle.

3D model produced with desmos, and exported using this project.