Topological dimension

We think of a point as 0-dimensional, a line as 1-dimensional, a plane as 2-dimensional, and in general euclidean space as n-dimensional. Intuitively, the dimension of the space equals the number of real parameters necessary to describe different points in the space. This intuitive view of dimension received two great jolts in the late nineteenth century, as we have already mentioned in this section:

• Cantor's proof that there is a one-to-one correspondence between and .
• Peano's construction of a continuous map from onto .

These results suggested that our intuitive notion of dimension was shaky, to say the least. It was a triumph of early 20th century mathematics to finally give a precise definition of topological dimension. This subject properly belongs near the end of a semester-long course on topology, but we would like to give the reader a flavor of the precision in topology as well as a glossary of common topological terminology.

Topology begins with the question of what it means for a set to be open or closed. In a metric space X this is easy to define.

open ball:

An open ball in X is a subset of the form

for given and radius . That is, contains all with distance from strictly less than .

open:

A subset is open if it is an arbitrary union of open balls in X. This means that every point in S is surrounded by an open ball which is entirely contained in X.

closed:

A subset is closed if its complement in X is open. This can also be described by saying that any point in X which is a limit of a sequence of points in S must also be contained in S.

In a topological space X, we do not assume that we have a distance function, but we do assume that we know what the open subsets are. This means that we furnish a family of subsets of X which we call the open subsets of X. This family should possess the three basic axioms of a topology:

• Both X and open.
• Any union of open sets is also open.
• Any intersection of only finitely many open sets is open.

If we require infinite intersections of open sets to be open, too many open sets which don't ``look open'' would have to be called open. For instance, in , any point is the intersection of infinitely many open balls with radius shrinking to 0. However, by the metric space definition, points are closed and not open. There are many loopholes in the above axioms, and addenda for closing them (and you wonder why mathematics is such a good training ground for the study of law?); however, these are generally the basic requirements for a topology. Commonly it is also assumed that points are closed, which is not strictly implied by the above axioms.

A subset S of a topological space X inherits a topology from X. We say is open if there is an open subset such that . This is called the subspace topology on S. There are two other closely related concepts.

covering:

A covering of a subset S is a collection of open subsets in X whose union contains all of S (and possibly more).

refinement:

A refinement of a covering of S is another covering of S such that each set B in is contained in some set A in . The idea is that the sets in are in some sense ``smaller'' than those in and provide a more finely detailed coverage of S.

Coverings play an important role in the definition of both topological and fractal dimension. In this figure, we show a covering of the Koch curve in red (the dotted lines indicate boundaries of open disks), and a refinement of in blue. Observe carefully how each blue disk is entirely contained in some red disk, and how the Koch curve belongs to the unions of both coverings.

  
Figure 37: Coverings of the Koch curve

We say a topological space X has topological dimension m if every covering of X has a refinement in which every point of X occurs in at most m+1 sets in , and m is the smallest such integer. Actually, this version of the definition of dimension (called the covering dimension) makes the most sense for compact spaces X. Our figure gives an illustration of finding a refinement of a covering of the Koch curve where each point of the curve belongs to at most two sets in the Koch curve, thereby illustrating why the Koch curve has topological dimension 1. We also show in this figure why a plane region such as a square is 2-dimensional: a refined covering has to have some triples of sets overlapping.

  
Figure 38: Coverings of the square

As an exercise, we encourage the reader to verify that the Cantor set is zero-dimensional.

The invariance of topological dimension and how this distinguishes between euclidean spaces and for we leave to the reader's future study and enjoyment of topology. Some references for both topological and fractal dimensions are [Edg90, Fal90].