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Intuitively, the dimension of a set, for example a subset of Euclidean space is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this naïve idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the cartesian coordinates of the point) , so in this sense, the plane is two-dimensional. As one would expect, topological dimension is always a natural number.
However, topological dimension behaves in quite unexpected ways on certain highly irregular sets such as fractals. For example the Cantor set has topological dimension zero, but in some sense it behaves as a higher dimensional space. Hausdorff dimension gives another way to define dimension, which takes the metric into account.
To define Hausdorff dimension for X, we first consider the number N(r) of balls of radius at most r required to cover X completely. Clearly, as r gets smaller N(r) gets larger. Very roughly, if the way that N(r) grows as r is squeezed down towards zero is a positive power d of r-1, then we say X has dimension d. In fact the rigorous definition of Hausdorff dimension is somewhat roundabout, since it first defines an entire family of covering measures for X. It turns out that Hausdorff dimension refines the concept of topological dimension and also relates it to other properties the space such as area or volume
It should be noted that there are various closely related notions of possibly fractional dimension. For example box-counting dimension, generalises the idea of counting the squares of graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller. In many cases these notions coincide, but the relation between them is highly technical.
The Hausdorff dimension (also: Hausdorff-Besicovitch dimension, capacity dimension and fractal dimension) was introduced by Felix Hausdorff. It gives an accurate way to measure the dimension of complicated sets such as fractals. The Hausdorff dimension agrees with the ordinary (topological) dimension on "well-behaved sets", but it is applicable to many more sets and is not always a natural number.
Suppose (X,d) is a metric space. We define a family of metric outer measureIn mathematics, in particular in measure theory, an outer measure is a function defined on all subsets of a given set with values in the extended real numbers. Outer measures are used to define measurable sets and countably additive measures. Measures ares on X using the Method II construction of outer measures due to Munroe and described in the article outer measureIn mathematics, in particular in measure theory, an outer measure is a function defined on all subsets of a given set with values in the extended real numbers. Outer measures are used to define measurable sets and countably additive measures. Measures are. Let C be the class of all subsets of X; for each positive real number s, let ps be the function A → diam(A)s on C. Hausdorff outer measure of dimension s, denoted Hs is the outer measure corresponding to the function ps on C.
Thus for any subset E of X
where the infimumIn mathematics the infimum of a subset of some set is the greatest element that is smaller than all other elements of the subset. Consequently the term greatest lower bound is also commonly used. Infima of real numbers are a common special case that is es is taken over sequences {Ai}i which cover E by sets each with diameter ≤ δ. Then
The value Hs(E) can be described directly as the infimum of all h > 0 such that for all δ > 0, E can be covered by countably many closed sets of diameter ≤ δ and the sum of the s-th powers of these diameters is less than or equal to h.
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