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...

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...

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

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...

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...

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...

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...

{\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)...

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...

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...

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...

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...

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...

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...

generic objects (intuitively objects that are somehow 'typical') by meeting dense sets. Both techniques are described as a relation (customarily denoted ⊩ {\displaystyle...

set in context of inclusion is smaller set in ordering and represents condition with more information). There are two types of important dense sets:...

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...

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...

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...

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...

is an alternative to polygonal modeling. An object is represented by a dense set of points or viewer-facing discs holding lighting information. Surfels...

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...

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...

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})...

of this curve, but not by compass and straightedge alone. Although a dense set of points on the curve can be constructed by compass and straightedge...