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		1.4.8— The Topological Dimension

		The topological dimension describes how the points that make up an object are connected together. The value of the topological dimension is always an integer. Edges, surfaces, and volumes have topological dimension of 1, 2, and 3.

		For example, we have already seen that the space filling properties of the perimeter of the Koch curve are described by the fractal dimension of about 1.2619. But no matter how wiggly this perimeter is, it is still a line. Thus the topological dimension of the perimeter of the Koch curve is equal to 1.

		There are a number of different ways to determine the topological dimension.

		1— Covering Dimension

		To evaluate the covering dimension we first find the least number of sets needed to cover all the parts of an object. These sets may need to overlap each other. If each point of the object is covered by no more than G sets, then the covering dimension d = G - 1.

		For example, if the least number of circles are used to cover a plane, each point in the plane will be covered by no more than 3 circles. Since 3 - 1 = 2, then the covering dimension of the plane is 2.

		2— Iterative Dimension

		The iterative dimension is based on the fact that a space of dimension D has borders that have dimension D-1. For example, a 3-dimensional volume can be surrounded by 2-dimensional planes. To evaluate the iterative dimension, we find how many times we need to take the borders of the borders of the borders . . . of an object to reach a zero-dimensional point. If we need to repeat the border taking H times, then the iterative dimension is d = H.

		For example, a plane can be surrounded by borders that are lines. The endpoints of the lines can be delimited by points. We need to take the borders of the borders twice to reach the points. Thus the iterative dimension of the plane is 2.

 

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