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Subset whose closure is the whole space
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
Dense_set
Mathematical set whose closure has empty interior
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
Nowhere_dense_set
Set of real numbers in mathematics
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
Smith–Volterra–Cantor_set
Does the plane contains a dense set of points whose distances are all rational
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
Erdős–Ulam_problem
Fractal named after mathematician Benoit Mandelbrot
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
Mandelbrot_set
Topological subset with no isolated point
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
Dense-in-itself
Type of ordering of a set
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
Dense_order
Well-spaced set of points in a metric space
discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points
Delone_set
"Small" subset of a topological space
set (also called a meager set or a set of first category) is a subset of a topological space that is a countable union of subsets that are not dense in
Meagre_set
Fractal sets in complex dynamics of mathematics
{\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)
Julia_set
Countable intersection of open sets
{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
Gδ_set
Concept in topology
(or first category) set (namely, a set that is a countable union of sets whose closure has empty interior, i.e., nowhere dense sets) and a nonmeagre (or
Baire_space
Graph with almost the max amount of edges
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected
Dense_graph
All points in the topological closure not belonging to the interior
1]} These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. They also show that it is possible
Boundary_(topology)
Volume rendering technique
difference to create a dense set of 3D Gaussians that represent the scene as accurately as possible.[citation needed] An optimized set of 3D Gaussians is
Gaussian_splatting
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
Glossary_of_general_topology
Uniqueness of countable dense linear orders
Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear orders are order-isomorphic. The theorem is named after
Cantor's_isomorphism_theorem
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
Meyer_set
On topological spaces where the intersection of countably many dense open sets is dense
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
Baire_category_theorem
Topics referred to by the same term
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
Density_(disambiguation)
Mathematical analysis of discontinuous points
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
Classification of discontinuities
Classification_of_discontinuities
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)
List_of_order_theory_topics
Class of sequences of natural numbers
constructions of dense Sidon sets involve the primes". A recent result of Balasubramanian and Dutta shows that if a dense Sidon set A = { a 1 , … , a
Sidon_sequence
French mathematician (1874–1932)
studied a combination of set theory and analysis topics to arrive at the Baire category theorem and the definition of a nowhere dense set. He then used these
René-Louis_Baire
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
Glossary_of_set_theory
Matrix in which most of the elements are zero
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
Sparse_matrix
Set of distances defined from a set of points
Erdős–Ulam problem asks whether it is possible to have a dense set in the Euclidean plane whose distance set consists only of rational numbers. Again, it remains
Distance_set
number field K {\displaystyle K} are not a dense set in Zariski topology. Erdős–Ulam problem: Is there a dense set of points in the plane all at rational
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Order-preserving mathematical function
{\displaystyle I} . Furthermore, the set A = { x ∈ I : f ′ ( x ) > 0 } {\displaystyle A=\{x\in I:f'(x)>0\}} is dense with positive Lebesgue measure. Not
Monotonic_function
Set of points on a line segment with certain topological properties
perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped
Cantor_set
Concept in mathematics
this case, structurally stable systems are typical, they form an open dense set in the space of all systems endowed with appropriate topology. In higher
Structural_stability
Function that is discontinuous at rationals and continuous at irrationals
maximum at each rational number, providing an example of a function with a dense set of proper local maxima. See the proofs for continuity and discontinuity
Thomae's_function
Fractal creation method
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
Chaos_game
Property holding for typical examples
holds on a dense open set, or more generally on a residual set, with the dual concept being a nowhere dense set, or more generally a meagre set. There are
Generic_property
Geometric theorem
point is a dense set in S2.) The axiom of choice can be used to pick exactly one point from every orbit; collect these points into a set M. The action
Banach–Tarski_paradox
Technique invented by Paul Cohen for proving consistency and independence results
\beta }} is dense, and a generic condition in it proves that the αth new set disagrees somewhere with the β {\displaystyle \beta } th new set. This is not
Forcing_(mathematics)
Method using forcing to construct sets with desired properties in computability theory
generic objects (intuitively objects that are somehow 'typical') by meeting dense sets. Both techniques are described as a relation (customarily denoted ⊩ {\displaystyle
Forcing_(computability)
Nowhere dense set Null set, conull set Partition regular Piecewise syndetic set Schnirelmann density Small set (combinatorics) Stationary set Syndetic set Thick
List of exceptional set concepts
List_of_exceptional_set_concepts
dense set Bounded set Totally bounded set Borel set Baire set Measurable set, Non-measurable set Universally measurable set Negligible set Null set Haar
List_of_types_of_sets
Concept in mathematical analysis
derivative of an everywhere differentiable function and that vanishes in a dense set. In particular, a Pompeiu derivative is discontinuous at every point where
Pompeiu_derivative
Vector space with a notion of nearness
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
Topological_vector_space
Classical problem in combinatorics
s^{\prime }} is the maximum cardinality set of S {\displaystyle S} . For δ − {\displaystyle \delta -} dense instances, however, there exists a c ln
Set_cover_problem
Unsolved conjecture in geometry
{\displaystyle k} -rational points of X {\displaystyle X} do not form a dense set in the Zariski topology on X {\displaystyle X} . The general form of the
Bombieri–Lang_conjecture
Branch of topology
Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty. Any open subspace of a Baire space is itself a Baire space.
General_topology
American mathematician
of Szemerédi's theorem on the existence of polynomial progressions in dense sets of integers. She is an associate professor in the department of mathematics
Sarah_Peluse
except possibly on a nowhere dense set, and the image of the medium under the mass density function being a bounded set) is negligible compared to the
Sound_particle
specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. If N is a dense submodule of M, it may alternatively
Dense_submodule
Method of drawing geometric objects
power of two and a set of distinct Fermat primes. In addition there is a dense set of constructible angles of infinite order. Given a set of points in the
Straightedge and compass construction
Straightedge_and_compass_construction
Topology on prime ideals and algebraic varieties
introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a
Zariski_topology
Curve where spinning and moving lines cross
they cannot be solved with compass and straightedge alone. Although a dense set of points on the curve can be constructed by compass and straightedge
Quadratrix_of_Hippias
On sets of points with integer distances
problem on the existence of dense point sets with rational distances. Although there can be no infinite non-collinear set of points with integer distances
Erdős–Anning_theorem
Type of plane curve
non-singular points form a dense set in the curve. It is also possible to construct convex curves for which the singular points are dense. A closed strictly convex
Convex_curve
strong measure zero set is countable, is independent of ZFC. A subset X of the real line is ℵ 1 {\displaystyle \aleph _{1}} -dense if every open interval
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Highly connected subgraph
In graph theory and computer science, a dense subgraph is a subgraph with many edges per vertex. This is formalized as follows: let G = (V, E) be an undirected
Dense_subgraph
Axiom in the mathematical field of set theory
satisfying the countable chain condition (hereafter ccc) and any set D = {Di}i∈I of dense subsets of P such that |D| ≤ κ, there is a filter F on P such that
Martin's_axiom
In abstract algebra, an E-dense semigroup (also called an E-inversive semigroup) is a semigroup in which every element a has at least one weak inverse
E-dense_semigroup
is an alternative to polygonal modeling. An object is represented by a dense set of points or viewer-facing discs holding lighting information. Surfels
Surfel
Indefinite integral
{\displaystyle g(x)=G'(x)=0} for all x in the set { F ( x n ) } n ≥ 1 {\displaystyle \{F(x_{n})\}_{n\geq 1}} which is dense in the interval [ F ( − 1 ) , F ( 1
Antiderivative
Field of mathematics and science based on non-linear systems and initial conditions
transitivity implies the existence of a dense set of points in X that have dense orbits. For a chaotic system to have dense periodic orbits means that every
Chaos_theory
Peptides released by neurons as intercellular messengers
are cleaved and post-translationally processed then packaged into large dense core vesicles. Neuropeptides are often co-released with other neuropeptides
Neuropeptide
Family of RISC-based computer architectures
in-depth knowledge gained from designing the instruction set enabled the code to be very dense, making ARM BBC BASIC an extremely good test for any ARM
Arm_architecture_family
American television sitcom (2007–2019)
as an official set by the Lego Ideas review board. On November 7, 2014, Lego Ideas approved the design and began refining it. The set was released in
The_Big_Bang_Theory
Linear operator on dense subset of its apparent domain
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function.
Densely_defined_operator
reduced form). (A dense set of discontinuities, namely the set of rational numbers.) The characteristic function of the Cantor set, which equals 1 if
Baire_function
Mathematical property of subsets in order theory
precisely the dense sets with respect to the right (respectively left) order topology. The cofinal relation over partially ordered sets ("posets") is
Cofinal_(mathematics)
Property of artificial neural networks
theorem formally states that a family of neural network functions is a dense set within a larger space of functions they are intended to approximate. In
Universal approximation theorem
Universal_approximation_theorem
Basic integral in elementary calculus
equivalent to IC is Riemann integrable: g, like IC, must be zero on a dense set, so as in the previous example, any Riemann sum of g has a refinement
Riemann_integral
German mathematician (1868–1942)
Hausdorff is also of fundamental importance: for each unbounded and ordered dense set A {\displaystyle A} there are two uniquely determined regular initial
Felix_Hausdorff
Role of coherent states
the existence of one such vector guarantees the existence of an entire dense set of such vectors in H {\displaystyle {\mathfrak {H}}} . Moreover, if the
Coherent states in mathematical physics
Coherent_states_in_mathematical_physics
Unsolved problem in mathematics
rational field Q {\displaystyle \mathbb {Q} } . In fact this set of rational numbers is dense in Q {\displaystyle \mathbb {Q} } . The discriminant of g(y
Inverse_Galois_problem
Subset of a preorder that contains all larger elements
Upper sets and lower sets are also known by many other names. An upper set may also be called an upward closed set, an up-set, an isotone set, or an
Upper_and_lower_sets
Theorem of dynamical systems
F n ≠ 0 {\displaystyle dF_{1}\wedge \cdots \wedge dF_{n}\neq 0} on a dense set Mutually Poisson commuting: the Poisson bracket { F i , F j } {\displaystyle
Liouville–Arnold_theorem
Construction in functional analysis, useful to solve differential equations
show this, we must show that Th − λ has dense range. Given f ∈ Lp(μ), again we consider the sequence of sets {Sn = h−1(B1/n(λ))}. Let gn be the characteristic
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
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
List of properties of sets of reals
List_of_properties_of_sets_of_reals
Earth, water, air, fire, and (later) aether
and mobile (εὐκινησιαν) while its opposite, earth, is blunt (αμβλυτητα), dense (παχυμερειαν), and immobile (ακινησιαν); they are joined by the intermediate
Classical_element
{\displaystyle V} of an abelian variety A {\displaystyle A} have a Zariski dense set of torsion points. The Manin-Mumford conjecture asserts that this occurs
Bogomolov_conjecture
nonempty open sets are disjoint. X cannot be written as the union of two proper closed subsets. Every nonempty open set is dense in X. Every open set is connected
Hyperconnected_space
Ordered chemical structure with no repeating pattern
of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors, which are the projections
Quasicrystal
Extends the Jordan curve theorem to characterize the inner and outer regions
Jordan-Schoenflies theorem can be proved as follows. The first step is to show that a dense set of points on the curve are accessible from the inside of the curve, i
Schoenflies_problem
Neural networks
Modern Hopfield networks (also known as Dense Associative Memories) are generalizations of the classical Hopfield networks that break the linear scaling
Modern_Hopfield_network
Wealthy people who travel widely for pleasure
The jet set is a social group of wealthy and fashionable people who travel the world to participate in social activities unavailable to ordinary people
Jet_set
Intersection of Set Theory and General Topology
satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P such that |D| ≤ k, there is a filter F on P such that F ∩ d is non-empty
Set-theoretic_topology
the particle trajectories. The ability to follow (track) a spatially dense set of individual particles for a sufficiently long period of time, and to
Particle_tracking_velocimetry
Set of real numbers that is not Lebesgue measurable
{Q} } partition R {\displaystyle \mathbb {R} } into disjoint sets, and each element is dense in R {\displaystyle \mathbb {R} } . Each element of R / Q {\displaystyle
Vitali_set
Set of data points in three-dimensional space
cloud is a discrete set of data points in space. The points may represent a 3D shape or object. Each point position has its set of Cartesian coordinates
Point_cloud
2002 video game
Set Radio Future is a 2002 action game developed by Smilebit and published by Sega for the Xbox. It is the sequel to the 2000 Dreamcast game Jet Set Radio
Jet_Set_Radio_Future
Signal processing method
using the equation given. Using Lagrange Interpolation, we compute the dense set of samples of A(ω) over the passband and stopband. Determine the new L+2
Parks–McClellan filter design algorithm
Parks–McClellan_filter_design_algorithm
Mathematical set with an ordering
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other.
Partially_ordered_set
category set) in only countably many points. Equivalently, A is an uncountable subset of R {\displaystyle \mathbb {R} } that meets every nowhere dense subset
Luzin_space
One-dimensional low-discrepancy sequence
\ldots .} The elements of the van der Corput sequence (in any base) form a dense set in the unit interval; that is, for any real number in [ 0 , 1 ] {\displaystyle
Van_der_Corput_sequence
Generalization of covers
small set. In topology, a map is a branched covering if it is a covering map everywhere except for a nowhere dense set known as the branch set. Examples
Branched_covering
Unit-distance-preserving maps are isometries
distinct images. The second function maps this dense set onto a countable unit simplex, an infinite set of points all at unit distance from each other
Beckman–Quarles_theorem
Special subset of a partially ordered set
collection of dense subsets (in the ordinal topology) of a, and ♣(a) meets each element of C. Replacing C with an arbitrary collection C̃ of dense sets, there
Filter_(mathematics)
French mathematician (1923–2002)
set of topologically stable mappings between two smooth manifolds is a dense set. Thom's lectures on the stability of differentiable mappings, given at
René_Thom
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
Fatou_conjecture
Condition of human breasts
Dense breast tissue, also known as dense breasts, is a condition of the breasts where a higher proportion of the breasts are made up of glandular tissue
Dense_breast_tissue
Root-finding algorithm
iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter. Fixed-point combinator Cobweb plot Markov chain Infinite
Fixed-point_iteration
Network that allows computers to share resources and communicate with each other
very low transmission loss and immunity to electrical interference. Using dense wave division multiplexing, optical fibers can simultaneously carry multiple
Computer_network
Motifs in the works of Jorge Luis Borges
Arena), he deals with another form of infinity; one whose elements are a dense set, that is, for any two elements, we can always find another between them
Jorge Luis Borges and mathematics
Jorge_Luis_Borges_and_mathematics
Network of weather stations based in the United States
self-contained weather sensors on unattended and wireless sites, allowing for a dense set of sample measurements to be collected. Mesonet Community Collaborative
Citizen Weather Observer Program
Citizen_Weather_Observer_Program
DENSE SET
DENSE SET
Boy/Male
Greek
God of wine.
Boy/Male
Australian, German, Greek, Hungarian
God of Wine; Wine; Drama; Follower of Dionysus
Girl/Female
French
Feminine of Denis from the Greek name Dionysus.
Surname or Lastname
English (Kent)
English (Kent) : variant spelling of Denn.
Girl/Female
American, Australian, British, English, Greek
Combination of Deana and Dina
Girl/Female
Christian & English(British/American/Australian)
Form of Dennis
Male
English
Variant spelling of English Dean, DENE means "dean, ecclesiastical supervisor."
Girl/Female
American, Australian, French, Greek
Mountain of Zeus; Feminine of Dennis; Follower of Dionysius
Female
English
Feminine form French Denis, DENISE means "follower of Dionysos."
Surname or Lastname
English
English : ethnic name for someone from Denmark, from Middle English den(s)ch ‘Danish’ (Old English denisc). There were many Danes in England in the Middle Ages, not only the long-established settlers in the Danelaw region, but also more recent immigrants.
Boy/Male
Christian, Indian
From Dionysisu; God of Wine
Girl/Female
Australian, Greek
The Mythological Mother of Perseus by Zeus; Form of Danae
Girl/Female
Indian
Boy/Male
English
From the valley.
Girl/Female
English
Combination of Deana (divine) and Dina (from the valley; avenged).
Girl/Female
English American French
From the Latin Dionysos or Dionysus, referring to the Greek god of wine.
Girl/Female
American, Australian, British, Chinese, Christian, Danish, Dutch, English, French, German, Greek, Indian, Portuguese, Swedish, Swiss
God of Wine; Feminine of Dennis; To Advise; Alone; Nun; Solitary; Follower of Dionysus
Girl/Female
Hindu, Indian
People who Give
Boy/Male
American, Anglo, Australian, British, Christian, English
From the Valley; Place Name; Valley; Occupational Name; Church Official
Girl/Female
Greek
Form of Danae; the mythological mother of Perseus by Zeus.
DENSE SET
DENSE SET
Girl/Female
Gujarati, Hindu, Indian, Kannada
Most Powerful
Boy/Male
Tamil
Shivas son Murugan, Well starred
Boy/Male
Tamil
Unique, One, United
Boy/Male
Arabic, Muslim
Fragrant; Sweet-smelling
Girl/Female
Norse
The goat who supplies mead for the gods.
Girl/Female
Indian, Punjabi, Sikh
Trust; Belief
Boy/Male
Ukrainian
God's gift.
Girl/Female
Spanish
sorrows.
Boy/Male
Afghan, Arabic
Servant of the Giver
Boy/Male
British, English, Irish
Rambunctious; Boisterous; High-spirited; Variant of Shaun from John
DENSE SET
DENSE SET
DENSE SET
DENSE SET
DENSE SET
v. t.
Perception by the sensory organs of the body; sensation; sensibility; feeling.
v. t.
Meaning; import; signification; as, the true sense of words or phrases; the sense of a remark.
v. t.
To perfume with odors from burning gums and spices.
v. t.
Perception through the intellect; apprehension; recognition; understanding; discernment; appreciation.
n.
One of the forms which a verb takes by inflection or by adding auxiliary words, so as to indicate the time of the action or event signified; the modification which verbs undergo for the indication of time.
a.
Having the constituent parts massed or crowded together; close; compact; thick; containing much matter in a small space; heavy; opaque; as, a dense crowd; a dense forest; a dense fog.
a.
Stupid; gross; crass; as, dense ignorance.
v. t.
Sound perception and reasoning; correct judgment; good mental capacity; understanding; also, that which is sound, true, or reasonable; rational meaning.
n.
A census; -- also, a public rate or tax.
v. t.
To grace.
n.
Manliness; dignity; comeliness; civility.
v. t.
Moral perception or appreciation.
v. t.
A faculty, possessed by animals, of perceiving external objects by means of impressions made upon certain organs (sensory or sense organs) of the body, or of perceiving changes in the condition of the body; as, the senses of sight, smell, hearing, taste, and touch. See Muscular sense, under Muscular, and Temperature sense, under Temperature.
v. t.
One of two opposite directions in which a line, surface, or volume, may be supposed to be described by the motion of a point, line, or surface.
n.
Condition; rank.
a.
Stretched tightly; strained to stiffness; rigid; not lax; as, a tense fiber.
v. t.
To perceive by the senses; to recognize.
v. t.
That which is felt or is held as a sentiment, view, or opinion; judgment; notion; opinion.
v. i.
To burn or scatter incense.