Many, if not most, objects in the observable universe exhibit some type of self-similarity. In particular, the objects show the same statistical and/or geometrical properties at many or all scales, where scale is determined by the observers measuring apparatus. Self-similarity is also a typical and defining property of the mathematical object known as a fractal.

In the past two articles we introduced the notion of a compact topological space. We need the extra structure of compactness to ensure that we cannot escape either off to infinity or outside the space itself along any sort of path that might be defined from within the space. Furthermore, if we manage to define and equip a suitable metric to our Fractal Manifold our space immediately becomes complete and totally bounded, which will be useful.

It is here that we take our first deviation from the beaten path. What I want to do now it add a little bit of structure to our compact topological space. I want to define a finite set which stands to index another set of non-surjective homeomorphisms from the space onto itself such that

$X=\bigcup_{j\in\mathcal{J}} f_j(X)$

This structure is going to keep track of how points on the compact topological space change under iteration of the space.