3.2 Topological Dimension
By definition, the null set (∅) and only the null set shall have the dimension −1. The dimension on any other space will be defined as one greater that the dimension of the object that could be used to completely separate any part of the first space from the rest. It takes nothing to separate one part of a countable set from the rest of the set. Since nothing (∅) has dimension −1, any countable set has a dimension of 0 (−1 + 1 = 0). Likewise, a line has dimension 1 since it can be separated by a point (0 + 1 = 1), a plane has dimension 2 since it can be separated by a line (1 + 1 = 2), and a volume has dimension 3 since it can be separated by a plane (2 + 1 = 3). We have to modify this dimension a little bit, however.
Sure a countable set can be separated by nothing, but it can also be separated by another countable set or a line or a plane. Take the rational numbers, for example. They form a countable infinite set. By embedding the set in the real number line, we could separate one point from any other with any single number. This set is has dimension 0, which would give the rational numbers a dimension of 1 (0 + 1 = 1). By embedding the set in the coordinate plane, we could also use any line with an x-intercept. This would give the rational numbers a dimension of 2 (1 + 1 = 2). We could also use planes if we embedded the set in a euclidean three-space and so on. I think it would be all right if we used the minimum value and called it the dimension of the space.
What about our composite spaces (╳) and ( · █)? We want the first to have dimension 1 and the second dimension 2. The x-shaped space is no problem. The least dimensional entity needed to separate it would be a point even at the intersection. The point and filled square is a bit more challenging. We need to distinguish between local dimension and global dimension. If we use the last definition and apply it to the set as a whole, then the space ( · █) would have dimension 0. If on the other hand, we examine it region by region we find that the point part has dimension 0 while any part of the square region has dimension 2. This is an example of a local dimension. The global dimension of the whole space should be two-dimensional so we need to modify our definition slightly. The dimension of a space should be the maximum of its local dimensions where the local dimension is defined as one more than the dimension of the lowest dimensional object with the capacity to separate any neighborhood of the space into two parts.
The measure defined above is called the topological dimension of a space. A topological property of an entity is one that remains invariant under continuous, one-to-one transformations or homeomorphisms. A homeomorphism can best be envisioned as the smooth deformation of one space into another without tearing, puncturing, or welding it. Throughout such processes, the topological dimension does not change. A sphere is topologically equivalent to a cube since one can be deformed into the other in such a manner. Similarly, a line segment can be pinched and stretched repeatedly until it has lost all its straightness, but it will still have a topological dimension of 1. Take the example below.
- Start with a line segment. Divide it into thirds. Place the vertex of an equilateral triangle in the middle third.
- Copy the whole curve and reduce it to ⅓ its original size. Place these reduced curves in place of the sides of the previous curve.
- Return to step 2 and repeat.
The result the is the Koch coastline, which evolves something like this.
With each iteration the curve length increases by the factor 4/3. The infinite repeat of this procedure sends the length off to infinity. The area under the curve, on the other hand, is given by the series
1+ (4/9) + (4/9)2 + (4/9)3 + …
which converges to 9/5 (assuming the area under the first curve is 1). These results are unusual but not disturbing. Such is not the case for the next curve.
- Take a unit square. Call it a cell.
- Divide each cell into four identical miniature copies of the original cell
- Draw a line starting in one cell so that it passes through every other cell until it returns to the starting position. (Also make sure the line does not stray too far from the previous iteration of the curve.)
- Return to step 2 and repeat.
The result is something like the diagrams below. (Cell lines were omitted in the third iteration for clarity. The last diagram represents the hypothetical result of an infinite iteration.)
This curve twists so much that it has infinite length. More remarkable is that it will ultimately visit every point in the unit square. Thus, there exists a continuous, one-to-one mapping from the points in the unit interval to the points in the unit plane. In other words, an object with topological dimension one can be transformed into an object with topological dimension two through a procedure that should not allow for such an occurrence. Simple bending and stretching should leave the topological dimension unchanged.
This is a Peano monster curve (actually, a variation on Hilbert's version of Peano's original), so called because of its monstrous or pathological nature. Since there are no such things as monsters, we have nothing to fear. The Koch and Peano curves raise questions about the meaning of dimension that will be answered in the next section.