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...

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...

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...

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...

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...

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...

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...

{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...

{\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)...

manifold Tensor density in differential geometry Dense set and nowhere dense set Dense-in-itself is a set that contains no isolated points Density (graph...

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...

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

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...

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...

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...

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)...

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...

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...

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})...

nowhere-dense sets (where a set is nowhere-dense if it is not dense in any open set). Then the negligible sets form a sigma-ideal. Let X be a directed set, and...

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

s^{\prime }} is the maximum cardinality set of S {\displaystyle S} . For δ − {\displaystyle \delta -} dense instances, however, there exists a c ln ⁡...

perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped...

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...

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...

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...

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...