Suppose we want to cover this curve with a series of open (in the sense of not having a precisely-defined boundary) disks. There are many different ways we could do it, three of which are shown in the figure above. In the case on the bottom left, several points are contained in the intersection of four disks; in the case in the middle, no point is contained in the intersection of more than three disks; finally, the case on the right leaves no point contained in the intersection of more than two disks. It’s easy to see that this is the furthest we could possibly push this covering: it wouldn’t be possible to arrange open disks of any size into any configuration where the curve was both completely covered and no disks overlapped[1]. We can use this to define topological dimensionality in general: for a given figure F, the topological dimension is defined to be the minimum value of n, such that every finite open cover of F has a finite open refinement in which no point is included in more than n+1 elements. In plain English, that just means that the topological dimension of a figure is one less than the largest number of intersecting covers (disks, in our example) in the most efficient scheme to cover the whole figure. Since the most efficient refinement of the cover for the curve above is one where there is a maximum of two disks intersecting on a given point, this definition tells us that the figure is 1-dimensional. So far so good—it’s a line, and so in this case topological dimensionality concurs with intuitive dimensionality[2].
There’s one more mathematical notion that we need to examine before we can get to the punch-line of this discussion: fractal dimensionality. Again, a simple example[3] can illustrate
- ↑ Why not? Remember that the disks are open, so points just at the “boundary” are not contained in the disks. Thus, a series of very small disks that were very near each other without intersecting would necessarily leave at least some points uncovered: those in the tiny region between two open disks. The only way to cover the whole figure is to allow the disks to overlap slightly.
- ↑ This also lets us move beyond our problem case from above: we can say why it is that a curve on a plane can be one-dimensional even though it is embedded in a two-dimensional space.
- ↑ This exceedingly clear way of illustrating the point is due to Mitchell (op. cit), though our discussion here is
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