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In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. [2] For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3.
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." [1] Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension.
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure , named for Felix Hausdorff , that assigns a number in [0,∞] to each set in R n {\displaystyle \mathbb {R} ^{n}} or, more generally ...
The Hausdorff dimension is useful for studying structurally complicated sets, especially fractals. The Hausdorff dimension is defined for all metric spaces and, unlike the dimensions considered above, can also have non-integer real values. [6] The box dimension or Minkowski dimension is a variant of the same idea.
(A completely regular space is Hausdorff if and only if it is T 0, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff. Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity. T 4 or Normal Hausdorff. A normal space is Hausdorff if and only if ...
Felix Hausdorff (/ ˈ h aʊ s d ɔːr f / HOWS-dorf, / ˈ h aʊ z d ɔːr f / HOWZ-dorf; [1] November 8, 1868 – January 26, 1942 [2]) was a German mathematician, pseudonym Paul Mongré (à mon gré (Fr.) = "according to my taste"), [3] who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and ...
Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space. Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers where = {,} is given the discrete topology.
A Hausdorff space, when used as an adjective, as in "the real line is Hausdorff" Hausdorff dimension, a measure theoretic concept of dimension; Hausdorff distance or Hausdorff metric, which measures how far two compact non-empty subsets of a metric space are from each other; Hausdorff density; Hausdorff maximal principle; Hausdorff measure