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Topological subset with no isolated point
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
Dense-in-itself
Subset whose closure is the whole space
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
Dense_set
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
Scattered_space
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
Small-scale current of rising air
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
Vertical_draft
Point of a subset S around which there are no other points of S
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).
Isolated_point
Subset that is closed and has no isolated points
X has cardinality at least 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} . Dense-in-itself Finite intersection property Subspace topology Jech, Thomas (2003)
Perfect_set
Hierarchical organization
– making the network denser and non-hierarchical. Because there is no highest level within a heterarchy but the heterarchy itself, control is decentralized
Dense_heterarchy
Set of all limit points of a set
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
Derived_set_(mathematics)
Embedding of Cantor set in 3-dimensional Euclidean space
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
Antoine's_necklace
Genre of alternative rock music
reached its peak in November 1991 with the release of My Bloody Valentine's Loveless. Noted for its innovative production and dense, immersive sound,
Shoegaze
On topological spaces where the intersection of countably many dense open sets is dense
singleton { x } {\displaystyle \{x\}} in X {\displaystyle X} is nowhere dense, and X {\displaystyle X} is meagre in itself.) In particular, this proves that the
Baire_category_theorem
Something that has mass and volume
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
Matter
"Small" subset of a topological space
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
Meagre_set
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
Glossary_of_general_topology
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
Pseudocomplement
Property of quark matter
quark matter is formed. It is expected that dense quark matter at low temperatures is a color superconductor. In the case of the SU(3) color group, a Bose–Einstein
SU(2)_color_superconductivity
Topology where a set is open if it contains a particular point
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
Particular_point_topology
Mathematical set whose closure has empty interior
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
Nowhere_dense_set
Island country in the central Pacific
tooth-embedded wooden spears, knives, and swords, and garbs of armour fashioned from dense coconut fibre. They were used instead of the gunpowder and weapons of steel
Kiribati
Capital of France
752 in 2023, 19.6% of the population of France, and the largest metropolitan area in the Eurozone. In 2012, the Commune of Paris was the most densely populated
Paris
Topological space with a dense countable subset
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
Separable_space
Linear operator defined on a dense linear subspace
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
Unbounded_operator
Hypothetical invisible cosmic material
that formed soon after the Big Bang. In the inflationary era and early radiation-dominated universe, extremely dense pockets of subatomic matter may have
Dark_matter
Capital and largest city of Egypt
thoroughfares. Areas around Tahrir Square, Ramsses, and Ataba are congested due to dense pedestrian activity. The Cairo Metro is a faster alternative to bypass street-level
Cairo
Certain topology in mathematics
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
Order_topology
Country in South Asia
area. Moderately dense forest, whose canopy density is between 40% and 70%, occupies 9.39% of India's land area. It predominates in the temperate coniferous
India
U.S. state
peak in the contiguous 48 states. Western Washington also is home of the Olympic Mountains, far west on the Olympic Peninsula, which support dense forests
Washington_(state)
Tabular arrangement of the chemical elements
They are brittle when solid as their atoms are held tightly in place. They are less dense and conduct electricity poorly, because there are no mobile
Periodic_table
Impassably dense forest, typically tropical
A jungle is land covered with dense forest and tangled vegetation, usually in tropical climates. Application of the term has varied greatly during the
Jungle
Capital and largest city of Bulgaria
neighbourhoods constructed after 2000 are densely built up and lack green spaces. There are four principal parks – Borisova gradina in the city centre and the Southern
Sofia
Destruction of objects by self-collapsing
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
Implosion (mechanical process)
Implosion_(mechanical_process)
Capital of Zhejiang Province, China
total GDP. The HHTZ positions itself as the "Silicon Valley" of China. The Alibaba Group is headquartered in the zone. In 2023 Alibaba sped up construction
Hangzhou
Type of dense black/brown hardwood
Ebony is a dense black–brown hardwood, coming from several species in the genus Diospyros, which also includes the persimmon. A few Diospyros species,
Ebony
Data structure for query optimization in databases
commonly used trees. A dense index in databases is a file with pairs of keys and pointers for every record in the data file. Every key in this file is associated
Database_index
Dog breed
(84 lb) in weight. Rhodesian Ridgebacks are typically very muscular and have a light wheaten to red wheaten coat, which should be short, dense, sleek and
Rhodesian_Ridgeback
Uniqueness of countable dense linear orders
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear
Cantor's_isomorphism_theorem
Religion from Japan
unaffiliated. They are found all over the country, from isolated rural areas to dense metropolitan ones. More specific terms are sometimes used for certain shrines
Shinto
Country in East Asia
Thus the habitable zones, mainly in the coastal areas, have very high population densities: Japan is the 40th most densely populated country even without
Japan
Birds from the order Strigiformes
is about 30 microseconds. Behind the ear openings are modified, dense feathers, densely packed to form a facial ruff, which creates an anterior-facing
Owl
Country in South America
deposed in April 1964 by a coup that resulted in a military dictatorship. The new regime was intended to be transitory, but gradually closed in on itself and
Brazil
Metaphysics concept covering the divide between two types of entities
which it is a part is in itself an abstract object. The abstract–concrete distinction is often introduced and initially understood in terms of paradigmatic
Abstract_and_concrete
Ongoing conflict since 2014
torture and rape of women and children, and many indiscriminate attacks in densely populated areas. After the Russian withdrawal from areas north of Kyiv
Russo-Ukrainian_war
Fastest known animal and common bird of prey
subspecies in the very dark, dense barring on its underside and its black ear coverts. Falco peregrinus furuitii, described by Momiyama in 1927, is found
Peregrine_falcon
Material with Vickers hardness exceeding 40 gigapascals
were synthesized by chemical vapor deposition in 1972. However, data on the attempted synthesis of B-C-N dense phases reported by different authors have been
Superhard_material
Type of cyclone occurring in the northern Atlantic coast of North America
the sea itself". His efforts, which included mailing hundreds of postcards, were profiled, just before his death in 2005 at the age of 88, in The New
Nor'easter
Capital and largest city of the Republic of Panama
a very dense skyline. There are more than 110 high-rise projects under construction, with 127 already built. The city holds the 45th place in the world
Panama_City
Country in West Asia
governed as an absolute monarchy under the House of Saud. In its Basic Law, Saudi Arabia defines itself as a sovereign Arab Islamic state with Islam as its
Saudi_Arabia
Preparing food using heat
oils. They are a major and dense source of food energy for many animals and play important structural and metabolic functions, in most living beings, including
Cooking
Capital and most populous city of Mexico
Overall precipitation is heavily concentrated in the summer months, and includes dense hail. Snow falls in the city scarcely, although somewhat more often
Mexico_City
Branch of statistics mathematics
field has focused more on applications and understanding the effects of dense and sparse observations schemes. The term "Functional Data Analysis" was
Functional_data_analysis
Removable disk storage medium
Katen A. (1996) [September 1992, April 1992]. Intel 82077SL for Super-Dense Floppies (PDF) (Application Note) (2 ed.). Intel Corporation, IMD Marketing
Floppy_disk
City in New York, United States
in Syracuse is in the two Valley neighborhoods, where 46.6% of the land is covered by trees. The lowest tree cover percentage is found in the densely
Syracuse,_New_York
American photographer and environmentalist (1902–1984)
calculate the proper exposure. In the resulting negative the foreground was underexposed, the highlights in the clouds were quite dense, and the negative proved
Ansel_Adams
Island in the southeastern Pacific
and porous tuff from Rano Raraku. A carving was abandoned when a large, dense and hard lithic fragment was encountered. However, these lithics became
Easter_Island
Isotope of hydrogen with one neutron
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)
Deuterium
Fast-moving current of hot gas and volcanic matter that moves away from a volcano
continuum of materials, spanning from denser, comparatively cooler rocks to warmer, lighter ashes and gasses. Denser rocks are more likely to be influenced
Pyroclastic_flow
Capital of Bavaria, Germany
people per km2, Munich is Germany's most densely populated municipality. It is also the second-largest city in the Bavarian dialect area after Vienna.
Munich
Star at the centre of the Solar System
matter gathered in the centre; the rest flattened into an orbiting disk that became the Solar System. The central mass became so hot and dense that it eventually
Sun
Charitable trust in India
residential buildings in densely populated areas of Mumbai. After the data was presented in court, permission to use private helipads in all urban areas of
Awaaz_Foundation
Borough and county in New York, US
second most densely populated county, and New York City borough, in the U.S. after Manhattan (New York County), and the most populous county in the state
Brooklyn
Chinese baked item
also appeared in Western countries as a form of delicacy. Most mooncakes consist of a thick, tender pastry skin enveloping a sweet, dense filling, which
Mooncake
Country in Central America
droughts in what is otherwise a seasonal desert is thought to have decimated the Maya, who relied on regular rainfall to support their dense population
Guatemala
Food made of flour and water
them may not hold its shape as the loaves rise, and their crumb may be dense with little aeration. Additives such as xanthan gum, guar gum, hydroxypropyl
Bread
2010 non-fiction book by Thilo Sarrazin
Germany Abolishes Itself: How We're Putting Our Country in Jeopardy (German title: Deutschland schafft sich ab: Wie wir unser Land aufs Spiel setzen) is
Germany_Abolishes_Itself
Genre of rock music
rhythms, dense bass-and-drum sound, and vigorous vocals. Heavy metal subgenres variously emphasize, alter, or omit one or more of these attributes. In a 1988
Heavy_metal_music
English rock band
bands who frequented the Syndrome club in London dubbed "The Scene That Celebrates Itself". NME magazine wrote in 1991, "[Blur] are [the] acceptable pretty
Blur_(band)
City in England
Manchester and Liverpool and below a handful of historic densely occupied, arguably overinflated markets in the local authorities: Harrogate, Cheltenham, Bath
Newcastle_upon_Tyne
Country within the United Kingdom
of state for transport. England has a dense and modern transportation infrastructure. There are many motorways in England, and many other trunk roads,
England
Country in Eastern Europe
populous country in Europe. It is a heavily urbanised country, and its industrial regions in the east and southeast are the most densely populated—about
Ukraine
Capital of the Philippines
per square kilometer (116,380/sq mi), Manila is one of the world's most densely populated cities proper. By 1258, a Tagalog-fortified polity called Maynila
Manila
Country in Northwestern Europe
population in 2021 was 56 million, representing some 84 per cent of the UK total. England is one of the most-densely-populated countries in the world,
United_Kingdom
Human-altered ecosystems
most prominent of which are listed below. Dense settlements are the second most densely populated regions in the world. They are defined as areas with
Anthropogenic_biome
Lithography using 13.5 nm UV light
In plasma-based EUV sources, ionization precedes much of the EUV emission. The thermal production of multicharged positive ions requires a hot, dense
EUV_lithography
State in central India
forest cover is less dense in the northern and western parts of the state, which contain the major urban centres. Variability in climatic and edaphic
Madhya_Pradesh
Country in West Asia
Mathieu, Edouard (6 September 2019). "Which countries are most densely populated?". Our World in Data. Archived from the original on 1 February 2022. Retrieved
Palestine
Country in Central America
total area of 21,041 km2 (8,124 sq mi). As the smallest and most densely populated country in continental America, El Salvador is affectionately called Pulgarcito
El_Salvador
U.S. state
several counties in Illinois, Indiana, and Wisconsin, and has a population of over 9.8 million. Chicago itself is a cosmopolitan city, densely populated, industrialized
Illinois
Federal capital district of the United States
operates the 446-acre (1.80 km2) United States National Arboretum, a dense arboretum in Northeast D.C. filled with gardens and trails. Its most notable landmark
Washington,_D.C.
Country in South America
the largest in the Americas. Most people live along the northern coast, centred on Paramaribo, making Suriname one of the least densely populated countries
Suriname
Largest city in New Zealand
is set to become even more densely populated in future which could ease the burden by creating higher density housing in the city centre. From around
Auckland
Country in Southeastern Europe
following clusters: the Argo-Saronic Islands in the Saronic gulf near Athens; the Cyclades, a large but dense collection occupying the central part of the
Greece
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
Family of birds
abruptly. In general, the aftershaft is absent; however, small ones on some tail and wing feathers may be present. Body feathers have very dense, fluffy
Columbidae
Interplay between observation, experiment, and theory in science
deduce that outer space was less dense than air, that is: "the body of the heavens is rarer than the body of air". In 1079 Ibn Mu'adh's Treatise On Twilight
Scientific_method
Large cat native to the Americas
and swamps with dense vegetation cover. In the Mayan forests of Mexico and Guatemala, 11 GPS-collared jaguars preferred undisturbed dense habitat away from
Jaguar
Country in Southern Africa
precipitation (50 mm (2.0 in) per year or less), frequent dense fog, and overall lower temperatures than in the rest of the country. In Winter, occasionally
Namibia
Capital of the Eastern Roman empire
the commercial areas facing the Golden Horn, had a dense population. Although the Genoese colony in Galata was small, it was overcrowded and had magnificent
Constantinople
Capital and most populous city of Finland
considered to be the only metropolis in Finland. It has a population of about 1.28 million, and is the most densely populated area of Finland. The Capital
Helsinki
Upper region of primate torso
"Menopausal Transition, Body Mass Index, and Prevalence of Mammographic Dense Breasts in Middle-Aged Women". Journal of Clinical Medicine. 9 (8): 2434. doi:10
Breast
Everything in space and time
starting from a dense fireball in an event called the Big Bang. The observable portion of the universe is approximately 93 billion light-years in diameter at
Universe
Class of animals with milk-producing glands
guard hairs measuring 30 cm (12 in) as well as a dense underfur, which forms an airtight coat, allowing them to survive in temperatures of −40 °C (−40 °F)
Mammal
Dense Pack. Dense pack itself was based on the dust defense concept, except in its basic form the dust was created by Soviet warheads attacking dense
Dust_defense
Air mass forming over a large body of water
cloud cover, or it may be accompanied by clouds. In many cases, marine layers can consist of dense fog. Often associated with marine layers are stratus
Marine_layer
Number divisible only by 1 and itself
involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory
Prime_number
National expressway in Germany
is too dense to justify driving only on the right lane. It is legal to give a short horn or light signal (flashing headlights or Lichthupe) in order to
Autobahn
Species of canine
recorded in Alaska and Canada. In central Russia, exceptionally large males can reach a weight of 69–79 kg (152–174 lb). The wolf has very dense and fluffy
Wolf
Capital and largest city of Germany
BCE dense human settlements along the Spree and Havel rivers gave rise to the Lusatian culture. Starting around 500 BCE Germanic tribes settled in a number
Berlin
Sub-region of the Asian continent
making it both the most populous and the most densely populated geographical region in the world. In 2022, South Asia had the world's largest populations
South_Asia
Province of Canada
Northumberland Strait. The island has two urban areas, and in total, is the most densely populated province in Canada. The island has a maritime climate, moderate
Prince_Edward_Island
travel, tourism, insurance
DENSE IN-ITSELF
DENSE IN-ITSELF
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DENSE IN-ITSELF
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DENSE IN-ITSELF
DENSE IN-ITSELF
DENSE IN-ITSELF
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