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 given in order to uniquely identify a point in that space. This definition is sufficient for most familiar spaces (such as all subsets of Euclidean spaces), but breaks down in the case of some more interesting figures. One of the cases in which this definition becomes fuzzy is the case of the Pythagoras Tree described above: because of the way the figure is structured, it behaves in some formal ways as a two-dimensional figure, and in other ways as a not two-dimensional figure.

The notion of topological dimensionality refines the intuitive concept of dimensionality. A full discussion of topological dimension is beyond the scope of this chapter, but the basics of the idea are easy enough to grasp. Topological dimensionality is also sometimes called “covering dimensionality,” since it is (among other things) a fact about how difficult it is to cover the figure in question with other overlapping figures, and how that covering can be done most efficiently. Consider the case of the following curve :

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