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A principle stating that in a metric space, every compact subset is closed and bounded. This principle is foundational in topology and related to the properties of Hausdorff spaces.

An alternate name for the Hausdorff dimension, recognizing the contributions of mathematician Abram Samoilovitch Besicovitch in the field of geometric measure theory.

A foundational result in fractal geometry which establishes that for any subset of a metric space, there exists a unique critical dimension at which the Hausdorff measure transitions from infinity to zero.

Packing measure is an alternative to Hausdorff measure that often behaves better on sets with a lot of overlapping structure, defined in a similar way to Hausdorff measure but using disjoint balls for the covering.

The Hausdorff measure is a measure that extends the concept of length, area, and volume to non-integer dimensions. It is defined for any non-negative real number and is a tool for analyzing the geometry of fractals.

A method for defining a measure, which can be used to define the Hausdorff measure. It involves covering a set with basic 'balls' whose diameters tend to zero and considering the infimum of the sum of a power of the diameters of these balls.

For a given dimension $s$, the $s$-dimensional Hausdorff measure is the measure obtained by the Carathéodory's Construction that corresponds to the $s$-dimensional volume of the set.

A lemma that provides conditions under which a set has positive measure, indicating that there exists a probability measure with certain decay properties on balls in a metric space.

A distance function defined between two subsets of a metric space, it is the greatest distance one must travel from a point in one set to reach the other set, and vice versa.