showing that dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open...
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topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered...
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X} will be a topological space. The definition of meagre set uses the notion of a nowhere dense subset of X , {\displaystyle X,} that is, a subset of X...
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Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it...
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the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff...
59 KB (7,732 words) - 19:34, 29 August 2024
dense-in-itself closed set is called a perfect set. (In other words, a perfect set is a closed set without isolated point.) The notion of dense set is...
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Every dense Gδ set in a Baire space is a Baire space. The result need not hold if the Gδ set is not dense. See the Examples section. Every comeagre set in...
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Unsolved problem in mathematics: Is there a dense set of points in the plane at rational distances from each other? (more unsolved problems in mathematics)...
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{\displaystyle \operatorname {J} (f)} is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers)...
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discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points...
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In mathematics, a partial order or total order < on a set X {\displaystyle X} is said to be dense if, for all x {\displaystyle x} and y {\displaystyle...
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Boundary (topology) (redirect from Boundary set)
Largest open set disjoint from some given set Interior (topology) – Largest open subset of some given set Nowhere dense set – Mathematical set whose closure...
21 KB (3,700 words) - 12:01, 6 July 2024
Density (disambiguation) (redirect from Dense (mathematics))
Look up density or dense in Wiktionary, the free dictionary. Density and dense usually refer to a measure of how much of some entity is within a fixed...
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it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the...
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Lindelöf but not separable. A space is separable if it has a countable dense set and hereditarily separable if every subspace is separable. It had been...
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Glossary of general topology (redirect from Locally-closed set)
intersection of any countable collection of dense open sets is dense; see Baire space. Baire space is the set of all functions from the natural numbers...
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however, do not necessarily consist of isolated points and may even be dense in an interval (a, b). For example, for any summable sequence ( a i ) (a_{i})...
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family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters. Furstenberg conjecture – is every invariant and ergodic...
190 KB (19,589 words) - 02:36, 29 August 2024
{Q} } were the intersection of open sets A n {\displaystyle A_{n}} each A n {\displaystyle A_{n}} would be dense in R {\displaystyle \mathbb {R} } because...
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Nowhere dense set Null set, conull set Partition regular Piecewise syndetic set Schnirelmann density Small set (combinatorics) Stationary set Syndetic set Thick...
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ternary construction only in passing, as an example of a perfect set that is nowhere dense (, Anmerkungen zu §10, /p. 590). More generally, in topology,...
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family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters. Świątek, Grzegorz; Graczyk, Jacek (1998), The real Fatou...
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Sparse matrix (redirect from Sparse and dense dimensions)
contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements...
25 KB (3,201 words) - 03:56, 29 August 2024
countable collection of open dense sets is dense basic set theory 1. Naive set theory 2. A weak set theory, given by Kripke–Platek set theory without the axiom...
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topological space such that the intersection of countably many dense open sets is still dense). It is used in the proof of results in many areas of analysis...
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union of compact convex sets is again compact and convex. Meager, nowhere dense, and Baire A disk in a TVS is not nowhere dense if and only if its closure...
103 KB (13,529 words) - 03:12, 5 July 2024
Cofinal and coinitial set, sometimes also called dense Meet-dense set and join-dense set Linked set (upwards and downwards) Directed set (upwards and downwards)...
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In mathematics, a Meyer set or almost lattice is a relatively dense set X of points in the Euclidean plane or a higher-dimensional Euclidean space such...
5 KB (576 words) - 11:58, 30 March 2023
property Universally Baire set Meager set Comeager set - A comeager set is one whose complement is meager. Null set Conull set Dense set Nowhere dense set...
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iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter. The "chaos game" method plots points in random order all...
14 KB (1,725 words) - 15:08, 26 July 2024