If *M* is a metric space, and *d* > 0 is a real number, then the *d*-dimensional **Hausdorff measure** *H*^{d}(*M*) is defined to be the infimum of all *m* > 0 such that for all *r* > 0, *M* can be covered by countably many closed sets of diameter < *r* and the sum of the *d*-th powers of these diameters is less than or equal to *m*.

It turns out that for most values of *d*, this measure *H*^{d}(*M*) is either 0 or ∞. If *d* is smaller than the "true dimension" of *M*, then *H*^{d}(*M*) = ∞; if it is bigger then *H*^{d}(*M*) = 0.

The **Hausdorff dimension** *d*(*M*) is then defined to be the "cutoff point", i.e. the infimum of all *d* > 0 such that *H*^{d}(*M*) = 0. The Hausdorff dimension is a well-defined real number for any metric space *M* and we always have 0 ≤ *d*(*M*) ≤ ∞.

- The Euclidean space
**R**^{n}has Hausdorff dimension*n*. - The circle S
^{1}has Hausdorff dimension 1. - Countable sets have Hausdorff dimension 0.
- Fractals are defined to be sets whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set (a zerodimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is ln(2)/ln(3) (see natural logarithm). The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2).