Hausdorff dimension

A numerical invariant of metric spaces, introduced by F. Hausdorff in [1]. Let be a metric space. For real and , let , where the lower bound is taken over all countable coverings of for which . The Hausdorff dimension of is defined as , where . The number thus defined depends on the metric on (on this, see also Metric dimension) and is, generally speaking, not an integer (for example, the Hausdorff dimension of the Cantor set is ). A topological invariant is, for example, the lower bound of the Hausdorff dimensions over all metrics on a given topological space ; when is compact, this invariant is the same as the Lebesgue dimension of .

References

[1]	F. Hausdorff, "Dimension and äusseres Mass" Math. Ann. , 79 (1918) pp. 157–179 MR1511917
[2]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)

Comments

The limit of the non-decreasing set is called the Hausdorff measure of in dimension . There is then a unique in the extended real line such that for and for . This real number is the Hausdorff dimension of . It is also called the Hausdorff–Besicovitch dimension.

See also (the editorial comments to) Hausdorff measure for more material and references. The Hausdorff dimension is a basic notion in the theory of fractals, cf. also [a1].

References

[a1]	K.J. Falconer, "The geometry of fractal sets" , Cambridge Univ. Press (1985) MR0867284 Zbl 0587.28004