|
In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is defined to be the minimum value of n, such that any open cover has a refinement in which no point is included in more than n+1 elements. In this context, a refinement is a second open cover such that every set of the second open cover is a subset of some set in the first open cover. It is named after Henri Lebesgue, although it was independently arrived at by a number of contemporaneous mathematicians. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset...
Henri Lebesgue Henri Léon Lebesgue (June 28, 1875, Beauvais – July 26, 1941, Paris) was a French mathematician, most famous for his theory of integration. ...
For example, consider some arbitrary open cover of the unit circle. This open cover will have a refinement consisting of a collection of open arcs. The circle has dimension 1, by this definition, because any such cover can be further refined to the stage where a given point x of the circle is contained in at most 2 arcs. That is, whatever collection of arcs we begin with, some can be discarded, such that the remainder still covers the circle, but with simple overlaps. Illustration of a unit circle. ...
Similarly, consider the unit disk in the two-dimensional plane. It is not hard to visualize that any open cover can be refined so that any point of the disk is contained in no more than three sets. In mathematics, the open unit disk around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: The closed unit disk around P is the set of points whose distance from P is less than or equal...
Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ...
The Lebesgue covering dimension coincides with the affine dimension of a finite simplicial complex; this is the Lebesgue covering theorem. In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ...
Some unusual topological constructions
The definition of the Lebesgue covering dimension can be used to build some unusual topological sets, such as the Sierpinski carpet. A construction can proceed as follows. Image File history File links Circle-question-red. ...
The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński. ...
Consider, for example, a finite open covering for the two-dimensional unit disk. This covering can always be refined so that no point in the disk belongs to more than three sets. Fixing this covering, remove all of the points in the disk that belong to three sets. Depending on the refinement, this will leave possibly one or more holes in the disk. The remaining object is again two-dimensional, and again has a finite open cover. The process of selecting a cover and refining, and then punching out holes can be repeated, ad infinitum. The resulting object is homeomorphic to the Sierpinski carpet. What is curious about this construction is that the carpet has a Lebesgue covering dimension of one, and not two. Given any open covering of the carpet, one can always find a refinement such that every point belongs to at most two sets. The proof of this is essentially by contradiction: were there a covering which required membership to three sets, then the affected area would have been punched out during the construction phase. As open covers are at most countable, every such case is handled during construction. Similar constructions can be performed in higher dimensions; the three-dimensional analogue is called the Menger sponge. Curiously, the Lebesgue covering dimension of the Menger sponge is again one. In mathematics, the open unit disk around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: The closed unit disk around P is the set of points whose distance from P is less than or equal...
Look up Ad infinitum in Wiktionary, the free dictionary. ...
This word should not be confused with homomorphism. ...
In mathematics, the Menger sponge is a fractal curve. ...
The Menger sponge has some additional curious properties. It is the universal curve. By this we mean that any possible one-dimensional curve (embedded in any number of dimensions) is homeomorphic to a subset of the Menger sponge. In a more restricted sense, any possible one-dimensional object embedded in the two-dimensional plane is homeomorphic to a subset of the Sierpinski carpet. Note that by curve we mean any object of Lebesgue dimension one; this includes trees and graphs with an arbitrary (countable) number of edges, vertices and closed loops. In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...
A labeled tree with 6 vertices and 5 edges In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. ...
History The idea of topological dimension first became a topic of considerable interest in the early 20th century. The core ideas were independently arrived at and published by Karl Menger, L. E. J. Brouwer, Pavel Urysohn and Henri Lebesgue. Karl Menger Karl Menger (Vienna, Austria, January 13, 1902 – Highland Park, Illinois, USA, October 5, 1985) was a mathematician of great scope and depth. ...
Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ...
Paul Samuilovich Urysohn (Па́вел Самуи́лович Урысо́н) (February 3, 1898 - August 17, 1924) was a Russian mathematician who is best known for his contributions in the theory of dimension, for...
Henri Lebesgue Henri Léon Lebesgue (June 28, 1875, Beauvais – July 26, 1941, Paris) was a French mathematician, most famous for his theory of integration. ...
See also 2-dimensional renderings (ie. ...
In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). ...
In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ...
References
Historical references - Karl Menger, General Spaces and Cartesian Spaces, (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley (1993) ISBN 0-201-58701-7
- Karl Menger, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.
Modern references - V.V. Fedorchuk, The Fundamentals of Dimension Theory, appearing in Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I, (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.
|
Feeds |
Contact