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 an irrational 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 homotopies. A homotopy 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.
The result the is so-called 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
which converges to 9/3 (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.
The result is something like the diagram 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 one-to-one mapping from the points on a line segment to the points in the unit plane. In other words, an object with topological dimension 1 can be transformed into an object with topological dimension 2 through a procedure that should not allow for such an occurrence. Simple bending and stretching should leave the topological dimension unchanged, however. This is the Peano monster curve. So called because of its monstrous or pathological nature. According to one source (Kline), we really have nothing to fear from the monster. The mapping from the line to the plane may be one-to-one but it is not continuous and is thus not a homotopy. This explains the change in dimension, but I am not satisfied. Kline never explains why the mapping is not continuous. I have never been happy with textbooks that leave the proof as an exercise for the reader.
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This page is a part of E-World! Maintained by Glenn Elert Last modified Saturday, 21-Feb-1998 15:58:28 EST Go to the nearest index |