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 |
Hausdorff dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hausdorff_dimension&oldid=28299