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...

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...

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...

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...

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...

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...

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...

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)...

{\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)...

discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points...

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...

Largest open set disjoint from some given set Interior (topology) – Largest open subset of some given set Nowhere dense set – Mathematical set whose closure...

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...

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...

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...

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...

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})...

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...

{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...

Nowhere dense set Null set, conull set Partition regular Piecewise syndetic set Schnirelmann density Small set (combinatorics) Stationary set Syndetic set Thick...

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,...

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...

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...

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...

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...

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...

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)...

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...

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...

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...