Hausdorff dimension

Summary

Let $A$ be a subset of a metric space. Its Hausdorff dimension is the exponent $s$ at which the $s$-dimensional Hausdorff measure of $A$ jumps from infinity to zero. It is a real number, not always an integer. It agrees with the usual dimension for smooth curves, surfaces and open sets, and it gives fractals such as the Cantor set non-integer dimensions.