In general topology, a subset A {\displaystyle A} of a topological space is said to be dense-in-itself or crowded if A {\displaystyle A} has no isolated...

is dense in X {\displaystyle X} if the smallest closed subset of X {\displaystyle X} containing A {\displaystyle A} is X {\displaystyle X} itself. The...

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...

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-in-itself. A set S {\displaystyle S} with S = S ′ {\displaystyle S=S'} is called a perfect set. Equivalently, a perfect set is a closed dense-in-itself...

X has cardinality at least 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} . Dense-in-itself Finite intersection property Subspace topology Jech, Thomas (2003)...

In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set...

typically be less dense than the surrounding region, and so will rise until it reaches air that is either warmer or less dense than itself. The converse will...

In mathematics, a scattered space is a topological space X that contains no nonempty dense-in-itself subset. Equivalently, every nonempty subset A of X...

the canonical example. A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set)....

must be single points. It is then easy to verify that A is closed, dense-in-itself, and totally disconnected, having the cardinality of the continuum...

requires it to take on a more dense form—in effect to be more concentrated, compressed, or converted into a denser material. In an implosion-type nuclear...

– making the network denser and non-hierarchical. Because there is no highest level within a heterarchy but the heterarchy itself, control is decentralized...

open set. Equivalently, a set is dense if its closure is the whole space. Dense-in-itself set A set is dense-in-itself if it has no isolated point. Density...

on the same set. No nonempty dense-in-itself subset Let S be a nonempty subset of X. If S contains p, then p is isolated in S (since it is an isolated point...

energy density and pressure to the vacuum itself. Fully 70% of the matter density in the universe appears to be in the form of dark energy. Twenty-six percent...

A jungle is land covered with dense forest and tangled vegetation, usually in tropical climates. Application of the term has varied greatly during the...

set is nowhere dense in R {\displaystyle \mathbb {R} } and hence meagre in R . {\displaystyle \mathbb {R} .} But it is nonmeagre in itself, since it is...

topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example...

be densely defined if its domain is dense in X. This also includes operators defined on the entire space X, since the whole space is dense in itself. The...

In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function....

Let M = Z \ {−1} = (0,1), then M is connected, so M is dense on itself and has no gaps, in regards to <. If −1 is not the smallest or the largest element...

Chicago by three horizontal white stripes. The North Side is the most-densely-populated residential section of the city, and many high-rises are located...

metres (33 ft) in length and thanks to its hollow stems, floats easily. This results in large clusters of stem amassing and create dense mats along the...

replace 1H in water molecules to form heavy water (2H2O), which is about 10.6% denser than normal water (so that ice made from it sinks in normal water)...

mask, dated to 9,000 BC, is attributed to the Maglemosian culture. In 2,000 BC dense human settlements along the Spree and Havel rivers gave rise to the...

132 in the 2022 census, it is the second least populous country in Asia and the ninth-smallest country by area, but also one of the most densely populated...

populous county in the state. Brooklyn, at 37,339.9 inhabitants per square mile (14,417.0/km2), is the second most densely populated county in the U.S. after...

x^{**}=1} ) is called dense. Every element of the form x ∨ x ∗ {\displaystyle x\vee x^{*}} is dense. D(L), the set of all the dense elements in L is a filter...

complements, denoted D(S) and called principal open sets, form the topology itself). This is the Zariski topology on A n . {\displaystyle \mathbb {A} ^{n}...