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mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties...

missing from the sequence, thereby invalidating the initial premise of countability. The contradiction is formed when defining this new real number, x {\displaystyle...

second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts...

Fréchet–Urysohn space. First-countability is strictly weaker than second-countability. Every second-countable space is first-countable, but any uncountable discrete...

1873. Weierstrass was first amazed by the concept of countability, but then found the countability of the set of real algebraic numbers useful. Cantor...

determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger...

resolved by noting that countability is not absolute to submodels of a particular model of ZFC. It is possible that a set X is countable in a model of set theory...

set does not necessarily exist in a countable model; that is, countability is "relative" to a model, and countable, first-order models are incomplete....

in Wiktionary, the free dictionary. In linguistics, a count noun (also countable noun) is a noun that can be modified by a quantity and that occurs in...

∅ ) = 0 {\displaystyle \mu (\varnothing )=0} Countable additivity (or σ-additivity): For all countable collections { E k } k = 1 ∞ {\displaystyle...

The cocountable topology, also known as the countable complement topology, is a topology that can be defined on any infinite set X {\displaystyle X} ...

ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions:...

π, e, and the golden ratio (φ). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are...

complement in X {\displaystyle X} is a countable set. In other words, Y {\displaystyle Y} contains all but countably many elements of X {\displaystyle X}...

(10 axioms and 1 schema) Axiom of Archimedes (real number) Axiom of countability (topology) Dirac–von Neumann axioms Fundamental axiom of analysis (real...

topological vector space (TVS) is said to be countably quasi-barrelled if every strongly bounded countable union of equicontinuous subsets of its continuous...

are countable. Proof of first theorem: If P(α) = ∅ for some index α, then P′ is the countable union of countable sets. Therefore, P′ is countable. The...

collection of things. One is most commonly a determiner used with singular countable nouns, as in one day at a time. The determiner has two senses: numerical...

X {\displaystyle X} is called countably generated if the topology of X {\displaystyle X} is determined by the countable sets in a similar way as the topology...

not absolute, because countability is based on equinumerosity and thus is not absolute. There may exist ordinals that are countable in V but uncountable...

present those entities. Many nouns have both countable and uncountable uses; for example, soda is countable in "give me three sodas", but uncountable in...

called an involution. Two is most commonly a determiner used with plural countable nouns, as in two days or I'll take these two. Two is a noun when it refers...

metrizable space, second-countability coincides with being Lindelöf, so X is second-countable. Conversely, if X is a Hausdorff second-countable manifold, it must...

in finitely many sets of the refining cover. A space is countably metacompact if every countable open cover has a point-finite open refinement. The following...

topological space is called countably compact if every countable open cover has a finite subcover. A topological space X is called countably compact if it satisfies...

can be expressed as the sum of five non-zero squares. There are five countably infinite Ramsey classes of permutations. 5 is conjectured to be the only...

thought of as synonymous with countability. Rather, boundedness is an underlying semantic distinction that motivates countability. Lexical aspect Grammatical...

a way that is compatible with countable unions and other kinds of countable limits of sets. For example, every countable subset of the real line has Lebesgue...

Generative Linguistics in the Old World. Keith Allan. 1980. Nouns and Countability. Language, 56(3):41–67. Chierchia, Gennaro (1998). "Reference to Kinds...

surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable base for the topology...

mathematics, the term countably generated can have several meanings: An algebraic structure (group, module, algebra) having countably many generators, see...

A supertask is an at least countably infinite sequence of operations that occur sequentially within a finite interval of time. Supertasks are called hypertasks...

In election science, a voting method satisfies the summability criterion if it is possible to tally election results locally by precinct, then calculate...

The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty...

space; i.e., a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively...

however, countably compact and thus not Lindelöf (a countably compact space is compact if and only if it is Lindelöf). In terms of axioms of countability, [...

is one of two closely related Boolean algebras, one countable and one complete. The countable Cantor algebra is the Boolean algebra of all clopen subsets...

In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step...

Löwenheim–Skolem theorem that it is not possible to characterize finiteness or countability, respectively, in first-order logic. Certain fragments of second-order...

a topological vector space (TVS) is said to be countably barrelled if every weakly bounded countable union of equicontinuous subsets of its continuous...

number of countable models of a first-order complete theory in a countable language is finite or ℵ0 or 2ℵ0. Morley showed that the number of countable models...

mathematics, a topological space is said to be σ-compact if it is the union of countably many compact subspaces. A space is said to be σ-locally compact if it...

each event depends only on the state attained in the previous event. A countably infinite sequence, in which the chain moves state at discrete time steps...

descriptive set theory, specifically invariant descriptive set theory, countable Borel relations are a class of relations between standard Borel space...

_{1}} are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So ℵ 1 {\displaystyle...

of subsets of X {\displaystyle X} closed under complement, countable unions, and countable intersections. The ordered pair ( X , Σ ) {\displaystyle (X...

a set of first category) is a subset of a topological space that is a countable union of subsets whose closures have empty interior. Thus meager sets...

rooms numbered 1, 2, 3, and so on with no upper limit. This is called a countably infinite number of rooms. Initially every room is occupied, and yet new...

In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear...

space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable space Lindelöf space Sigma-compact...

the sentence. The head of expressions of zero quantity can be either a countable or an uncountable noun. Linguistically, expressions that denote zero quantity...

measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. A null set is not...

is redundant in the classical context, but otherwise need not imply countability. The existence of injections from the uncountable 2 N {\displaystyle...

requirements: First, the probability of a countable union of mutually exclusive events must be equal to the countable sum of the probabilities of each of these...

situation came to be known as Skolem's paradox; it shows that the notion of countability is not absolute. For each first-order σ {\displaystyle \sigma } -formula...

"chad" has been used both as a mass noun (as in "a pile of chad") and as a countable noun (pluralizing as in "many chads"). The origin of the term chad is...

theorem in 1959, is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive...

air, information, physics. However, many nouns of this type also have countable meanings or other contexts in which a plural can be used; for example...

Countable Corporation (a.k.a. Countable) is a Software-as-a-Service (SaaS) company based in San Francisco. The company was founded in 2013 by its CEO...

infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence is postulated...

over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance of the notion comes...

The union of any countable family of countable sets is countable. (In ZF, this statement is logically implied by the axiom of countable choice.) Every infinite...

informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number:...

index set I {\displaystyle I} need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This...

ways: the first employs a translation-invariant metric, the second a countable family of seminorms. A topological vector space X {\displaystyle X} is...

This particular counterexample shows that second-countability does not follow from first-countability. Several other "Counterexamples in ..." books and...

set of events in a σ-algebra that satisfies measure properties such as countable additivity. The difference between a probability measure and the more...

most countable model. However, Cantor's theorem proves that there are uncountable sets. The root of this seeming paradox is that the countability or noncountability...

machine-checkable proof by construction of the existence of a non-sofic countable discrete group. The existence of such groups was previously an open question...

progression, is a standard trick in working with infinite sequences and countability. The algebraic rules of this diagonal-shaped function can verify its...

Banach and Tarski showed that an analogous statement remains true if countably many subsets are allowed. The difference between dimensions 1 and 2 on...

assumption of second-countability from the definition of a surface, there exist (necessarily non-compact) topological surfaces having no countable base for their...

must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that...

first-order logic is, up to equivalence, the only abstract logic that is countably compact and has Löwenheim number ω. Abstract algebraic logic – Aspect...

named after Wacław Sierpiński who proved it in 1920. It states that any countable metric space without isolated points is homeomorphic to Q {\displaystyle...

categoricity theorem is a theorem of stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical...

dimensionless. More specifically, counting numbers can be used to express countable quantities. The concept is formalized as quantity number of entities (symbol...

inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens...

space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle X} has a limit point...

prescriptivists usually say that fewer should only be used with plural countable nouns, and that less should be used only with uncountable nouns. This...

set function is a function that has the additivity property even for countably infinite many sets, that is, μ ( ⋃ n = 1 ∞ A n ) = ∑ n = 1 ∞ μ ( A n )...

second-countable space is paracompact. The Sorgenfrey line is paracompact, even though it is neither compact, locally compact, second countable, nor metrizable...

+ 2 ) {\displaystyle (n,n+2)} for an integer n {\displaystyle n} . A countable collection of subsets need not be locally finite, as shown by the collection...

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories...

as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883...

support from other users. In 2019, Causes was acquired by Countable Corporation. In 2020, Countable.us and its associated app merged with Causes.com. Causes...

{\displaystyle X} is metrizable if and only if it is regular and has a countably locally finite (that is, 𝜎-locally finite) basis. A topological space...

of T is the class of all cardinals κ such that T is stable in κ. For countable theories there are only four possible stability spectra. The corresponding...

transcendental numbers. Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves...

there must exist countable non-standard models of arithmetic. One way to define such a model is to use Henkin semantics. Any countable non-standard model...

compactness and sequential compactness are equivalent (using the axiom of countable choice). However, there exist sequentially compact topological spaces...

elements of X . {\displaystyle X.} If the complement is not finite, but is countable, then one says the set is cocountable. These arise naturally when generalizing...

(in a measure space), "finite" (when infinite sets are involved), or "countable" (when uncountably infinite sets are involved). For example: The set S...

In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. The inductive definition above is well-founded...

against founding mathematics on first order logic. He explained that the countability or uncountability of a set is not absolute, but relative to the model...

(this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in ℵ 0...

together with only the ultrafilter lemma: A countable union of countable sets is a countable set. The axiom of countable choice (ACC). The axiom of dependent...

if it has a countable dense subset. First-countable. A space is first-countable if every point has a countable local base. Second-countable. A space is...

[0,ω1] is not first-countable. The subspace [0,ω1) is first-countable however, since the only point in [0,ω1] without a countable local base is ω1. Some...

relation on a standard Borel space X is a Borel equivalence relation E with countable classes, that can, in a certain sense, be approximated by Borel equivalence...

mathematics, Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear orders are order-isomorphic. The theorem is named...

measure spaces by observing that the countable union of countable unions is again a countable union, and that the countable unions of null sets are null. A...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

space without isolated points in which every nowhere-dense subset is countable. There are many minor variations of this definition in use: the T1 condition...

The indefinite article is the default determiner for other singular, countable, common nouns, while no determiner is the default for other common nouns...

analog of Fubini's theorem for arbitrary second countable Baire spaces. Let X and Y be second countable Baire spaces (or, in particular, Polish spaces)...

equivalent (over ZF) conditions: it has a countably infinite subset; there exists an injective map from a countably infinite set (say, N, the set of all natural...

embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists...

subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having...

draw conclusions about the positive recurrence of Markov chains with countable state spaces. It uses the fact that positive recurrent Markov chains exhibit...

With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider the distinct...

object such as an integer, permutation or graph must be found from a countable set. A problem with continuous variables is known as a continuous optimization...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x {\displaystyle x} in a Hilbert...

isomorphism between countably infinite structures satisfying specified conditions. In particular it can be used to prove that: any two countably infinite densely...

discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed...

mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case...

confused with the countable ordinal obtained by ordinal exponentiation). The Baire space is defined to be the product of countably infinitely many copies...

all sets are contained in some single ultimate model. The collection of countable transitive models of ZFC (in some universe) is called the hyperverse and...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

Much was discussed at the meeting. more (also countable, plural) – More is better. most (also countable, plural) – Most was rotten. (Usually specified...

its subspace topology). The Cantor set is naturally homeomorphic to the countable product 2 _ N {\displaystyle {\underline {2}}^{\mathbb {N} }} of the discrete...

is called countably saturated if it is ℵ 1 {\displaystyle \aleph _{1}} -saturated; that is, it realizes all complete types over countable sets of parameters...

In theoretical computer science, a discrete system is a system with a countable number of states. Discrete systems may be contrasted with continuous systems...

Unsolved problem in mathematics Does every countable graph have an unfriendly partition into two parts? More unsolved problems in mathematics In the mathematics...

necessarily well-ordered by the lexicographical order. For instance, the set of countably infinite binary sequences (by definition, the set of functions from natural...

{\displaystyle \omega _{1}} are the countable ordinals, the supremum of every countable family of countable ordinals is a countable ordinal, and every increasing...

been almost completely solved for the case of a countable theory T. In this section T is a countable complete theory and κ is a cardinal. The Löwenheim–Skolem...

Character Groups, refined the topology on character groups and removed countability restrictions, bringing the theory close to its modern formulation. Claude...

rational greater than sup x. Uα + 1 is countable as it consists of countably many extensions of each of the countably many elements in Uα. If α is a limit...

{\displaystyle M} that is invisible within M {\displaystyle M} (e.g. the countability of M {\displaystyle M} ), and thus prove the existence of sets that are...

cover can be as small as that of any topological basis. Hence, second countability implies space is Lindelöf. A refinement of a cover C {\displaystyle C}...

an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical...

such objects; the class or instance may be an idea, person, physical countable object (or class thereof), or physical noncountable substance (or class...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

sequence of at most countable cardinalities is realized. Helmut Ulm (1933) found an extension of this classification theory to general countable p-groups: their...

uncountable cardinal is an admissible ordinal. By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar...

{\displaystyle \operatorname {cor} (X)} . Locally compact spaces with countable core generalize σ-compact locally compact spaces. The concept was introduced...

monotonic functions: An increasing function f on an interval I has at most countably many points of discontinuity. Let A n = { x ∈ I ; f n ( y ) ↛ f n ( x...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

rather than the academic field studying what happened. When used as a countable noun, a history is a representation of the past in the form of a history...

constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called Borel sets) plus-minus null...

set is sometimes used for countable sets. However it is also often used for computably enumerable sets, which are the countable sets for which an enumeration...

x}p(\omega ).} The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers. A discrete...

collection. A collection of subsets of X is said to be countably locally discrete, if it is the countable union of locally discrete collections. Locally discrete...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

a language is assumed to be a countable set of sentences, each of finite length, and constructed out of a countable set of symbols. A theory of syntax...

general topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German nouns...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that F is necessarily itself countable. Paul...

mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

also discrete space). Any discrete subset S of Euclidean space must be countable, since the isolation of each of its points together with the fact that...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

considered a separate process. list An abstract data type that represents a countable number of ordered values, where the same value may occur more than once...

orthocompact if and only if X is countably metacompact. (B.M. Scott) Every orthocompact space is countably orthocompact. Every countably orthocompact Lindelöf space...

is not necessarily finite, or even countable. The set of transitions is not necessarily finite, or even countable. No "start" state or "final" states...

More generally take X {\displaystyle X} and Y {\displaystyle Y} both countable and A = P ( X ) ,   B = P ( Y ) {\displaystyle {\mathcal {A}}={\mathcal...

ba(\Sigma )} consisting of countably additive measures. The notation ba is a mnemonic for bounded additive and ca is short for countably additive. If X is a...

types in PHP 8.1 (e.g. Traversable&Countable meaning the value must implement both the Traversable and Countable interfaces), and disjunctive normal...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

is known as its state space. In a discrete system, the state space is countable and often finite. The system's internal behaviour or interaction with...

hereditarily finite sets. Another example is the set of hereditarily countable sets. Admissible ordinal Barwise, Jon (1975). Admissible Sets and Structures:...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

supercompact space (or any continuous image of one), the cluster point of any countable subset is the limit of a nontrivial convergent sequence. de Groot (1969)...

the Haar measure. (This is a Borel regular measure when G is second-countable; the left and right Haar measures coincide when G is compact.) Consider...

likewise describes how mass nouns are understood when they are used as countable nouns. In this case, the plural noun may be understood as naming individual...

that property, then so does every subspace of it. For example, second-countability is a hereditary property. Homeomorphism If X and Y are spaces, a homeomorphism...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

applications Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. Because the...

with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

space. In terms of countability axioms, R l {\displaystyle \mathbb {R} _{l}} is first-countable and separable, but not second-countable. In terms of compactness...

theory, the Henson graph Gi is an undirected infinite graph, the unique countable homogeneous graph that does not contain an i-vertex clique but that does...

the compactness degree of the space X . {\displaystyle X.} Axioms of countability Lindelöf's lemma Steen & Seebach, p. 19 Willard, Def. 16.5, p. 110 Willard...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

topological space is also second-countable (there are only finitely many open sets) and separable (since the space itself is countable). If a finite topological...

is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

limited to material bodies. In this regard, the word thing is the broadest countable noun, reflecting the slogan "everything is a thing". However, the term...

a square lattice in the complex plane. The Eisenstein integers are a countably infinite set. The Eisenstein integers form a commutative ring Z[ω] of...

X ) {\displaystyle w(X)} . From the examples above, the real line has countable weight. If B {\displaystyle {\mathcal {B}}} is a base for the topology...

states that every Hausdorff second-countable regular space is metrizable. So, for example, every second-countable manifold is metrizable. (Historical...

endpoints is atomic. Any prime model of a countable theory is atomic by the omitting types theorem. Any countable atomic model is prime, but there are plenty...

of the family. A σ-locally finite or countably locally finite collection is a family that is the union of countably many locally finite families. A cover...

an analog of passing to subsequences.) For first countable spaces (meaning each point has a countable neighborhood base), sequences often suffice, in that...

' Many uncountable nouns, such as furniture and clothing, can be made countable. By feeling the cardboards, it can fold up to 600 cartons a day. – 'By...

paranormal space (Nyikos 1984) is a topological space in which every countable discrete collection of closed sets has a locally finite open expansion...

existence of probability measures for probabilistic events consisting of a countably infinite number of individual probabilistic events. In particular, the...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

> 0 {\displaystyle n>0} . The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal...

An axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain...

is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich...

mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers)...

{Q} /\mathbb {Z} \cong \sum _{p}\mathbb {Q} _{p}/\mathbb {Z} _{p}} . Countability assumption in the second Prüfer theorem cannot be removed: the torsion...

commonly used as a mass noun ("some Lego") or, in American English, as a countable noun with plural "Legos", to refer to the bricks themselves, (e.g. genericization)...

real closed fields Every uncountably categorical countable theory Every countably categorical countable theory A group of three elements True arithmetic...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

thought of as spaces that satisfy a very weak axiom of countability, and all first-countable spaces (notably metric spaces) are sequential. In any topological...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

closed category, a concept in category theory CCC, Roman numeral for 300 Countable chain condition, in order theory CCCn, cube-connected cycles of order...

Huntington develops the back-and-forth method to prove Cantor's result that countable dense linear orders (without endpoints) are isomorphic. 1908 – Ernst Zermelo...

Pips are small but easily countable items, such as the dots on dominoes and dice, or the symbols on a playing card that denote its suit and value. In...

determined by its values on a single open neighborhood in D, or even a countable subset of D with an accumulation point (provided this contains a converging...

show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step and if some of...

certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property. Lindström's...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

theorem, says: Every syntactically consistent, countable first-order theory has a finite or countable model. Given Henkin's theorem, the completeness...

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input...

natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that...

Macmillan Publishers Limited. 2010. Retrieved 2010-04-14. adapter noun [countable] /əˈdæptə(r)/ or adaptor an object that allows you to connect two pieces...

Morocco and abroad. Tarifiyt has two genders, masculine and feminine. In countable nouns with Berber affixes, gender is derivative: in principle, every masculine...

probability theory, the closure of a collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection...

probability distributions is a subset of the set of all programs, which is countable. Similarly, the sets of observable data considered by Solomonoff were...

intervention tends to be specific and concrete, and thus the word is often countable; for example, one instance of cardiac catheterization is one intervention...

Every countable formally real field is orderable. Uniqueness of algebraic closure (for a countable field). The De Bruijn–Erdős theorem for countable graphs:...

known about infinitely generated abelian groups, even in the torsion-free countable case. An abelian group ⟨ G , + , 0 ⟩ {\displaystyle \langle G,+,0\rangle...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

has no least element. However, since the set of rational numbers ℚ is countable, there exists a well ordering of ℚ that is order isomorphic to the standard...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

of multiplicity refers to parts which are quantitative, distinct, and countable, and the other form of multiplicity refers to parts that are qualitative...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

cumulative hierarchy is much weaker, as is shown by the existence of countable models (Skolem's paradox), but it enjoys some important advantages." In...

it is a probability space consisting of an interval and/or a finite or countable number of atoms. The theory of standard probability spaces was started...

box}}(A_{1}),\dots ,\dim _{\text{upper box}}(A_{n})\}.} However, it is not countably stable, i.e. this equality does not hold for an infinite sequence of sets...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not...

language over a finite or countably infinite alphabet Σ {\displaystyle \Sigma } is countable, since it is a subset of the countably infinite set Σ ∗ {\displaystyle...

closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying...

jump and removable discontinuities. f {\displaystyle f} can only have countably many discontinuities in its domain. The discontinuities, however, do not...

Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof...

of rational numbers is the field of algebraic numbers. There are many countable algebraically closed fields within the complex numbers, and strictly containing...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

freedom (physics and chemistry), a concept describing dependence on a countable set of parameters Degree of frost, a unit of temperature measurement Degrees...

continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select...

(generalizing a conjecture of Roland Fraïssé on countable orders) states that the embedding relation on the class of countable unions of scattered orders is a well-quasi-order...

that are differentiable on a cocountable set, as the image of a countable set is countable and thus a null set, but not to functions differentiable on a...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

of an event or entity. Many languages have classifiers that identify countable nouns as belonging to a particular type or having a particular shape....

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

applicant to own no more than $2,000 in countable assets and a married applicant to own no more than $3,000 in countable assets. Certain assets, such as the...

referring to groups of (usually) people. Both "team" and "government" are countable nouns (consider: "one team", "two teams", "most teams"; "one government"...

more general measure theory. This is the assumption of σ-additivity: Any countable sequence of disjoint events (synonymous with mutually exclusive events)...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or a clique with the same cardinality as the...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

natural numbers, the set of integers ⁠ Z {\displaystyle \mathbb {Z} } ⁠ is countably infinite. An integer may be regarded as a real number that can be written...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

in general), a space is cosmic if and only if it has a countable network; namely a countable collection of subsets of the space such that any open set...

A universe (mathematics) obtained from a set by taking the power set countably many times. A structure (mathematical logic) of which another structure...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

universe of all sets M is a countable transitive model of set theory G is a generic subset of P over M. P satisfies the countable chain condition if every...

are uncountable infinite sets, which are larger than countable ones; therefore, if only a countable infinity of bottles fall, an uncountable number remains...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation...

It also has a countable dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a countable dense subset and...

In October 2016, the ABA voted to add the U.S. state of Hawaii to the countable area for ABA Big Years. All the 2016 big year birders except Hagenlocher...

at most εn. Borel's conjecture, that every strong measure zero set is countable, is independent of ZFC. A subset X of the real line is ℵ 1 {\displaystyle...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

\sigma } -finite subset is a measurable subset which is the union of a countable number of measurable subsets of finite measure. The measure μ {\displaystyle...

mathematical constants such as the number π. The set of periods form a countable ring and bridge the gap between algebraic and transcendental numbers....

formulae are fit to discrete data—data that could potentially take on only a countable set of values, such as the integers, and which are not infinitely divisible...

In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

Primary maritime cargo types Cargo type Countable Packaging Container Remarks Break bulk cargo or general cargo Countable Yes No Break bulk cargo or general...

to determining if some functions are countable. For example, the set of all even integers f(n)=2n is countably infinite because there’s a bijection between...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

enumerated at 108 (an auspicious number in Hinduism and the number of countable beads on a japa mala, similar to a rosary, for repetition of mantras)...

assets and IP, including the Causes assets, were purchased by GovTech app Countable. James Windon was the president of Brigade. He previously acted as a vice-president...

algebra with at least λ elements but generated by a countable number of elements. As the size of countably generated complete Boolean algebras is unbounded...

empty set and the entire set X {\displaystyle X} , and is closed under countable union and complement. Then we can define the Borel σ-algebra over X {\displaystyle...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

algebra of all Baire sets modulo meager sets in a topological space with a countable base is complete; when the topological space is the real numbers the algebra...

Hamkins proved that any two countable models of set theory are comparable by embeddability, and in particular that every countable model of set theory embeds...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

= Π1 0 = Δ1 0 = boldface arithmetical ⋮ ⋮ Δ0 α (α recursive) Δ0 α (α countable) Σ0 α Π0 α Σ0 α Π0 α ⋮ ⋮ Σ0 ωCK 1 = Π0 ωCK 1 = Δ0 ωCK 1 = Δ1 1 = hyperarithmetical...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

components may be countably infinite in number. More general cases (i.e. an uncountable set of component distributions), as well as the countable case, are treated...

Professor Rocket gives the viewers long math problems for them to solve with countable pictures or descriptions of numbered things, such as the number of members...

of even a countable infinite set, that is, of a set whose cardinality is ℵ 0 {\displaystyle \aleph _{0}} . Even if GST did afford a countably infinite...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

\{\omega +n:n<\omega \}} viewed as subsets of B {\displaystyle B} have the countable cardinality of the cofinality of B {\displaystyle B} but are not order...

Turing machines that are capable of carrying out computations involving a countably infinite number of algorithmic steps. These machines are ruled out in...

theory context: Ω {\displaystyle \Omega } is the disjoint union of a countable family of sets in S . {\displaystyle {\mathcal {S}}.} A field of sets...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

uncountable ordinal) that are not expressible. Such ordinals are known as large countable ordinals. The operations of addition, multiplication and exponentiation...

Development (drafting), a type of technical drawing Development (topology), a countable collection of open coverings Development, a term used in chess Development...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

function whose set of nondifferentiable points is something other than a countable set of points. Analogous results for better behaved classes of continuous...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian...

Panará (panãra pẽẽ, pronounced [panə̃ˈɾa ˈpẽj̃]), also known as Kreen Akarore (from Mẽbêngôkre Krã jakàràre, pronounced [ˈkɾʌ̃ jaˈkʌɾʌɾɛ]) and spelled...

come up with a method of grouping the microstates together to obtain a countable set. This procedure is known as coarse graining. In the case of the ideal...

property of being normal. Every locally compact group which is T0 and first-countable is metrisable as a topological group (i.e. can be given a left-invariant...

Specifically, if a continuous function F(x) admits a derivative f(x) at all but countably many points, then f(x) is Henstock–Kurzweil integrable and F(b) − F(a)...

with these themes. The first dealt with partitioning an interval into countably many congruent subsets. It solved a problem of Hugo Steinhaus asking whether...

{\displaystyle a} that are real numbers are also algebraic. There are only countably many algebraic numbers, but there are uncountably many real numbers, so...

continuous. X is path-connected and so connected. X is second-countable, and therefore is first-countable, separable and Lindelöf. All subspaces of X have the...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

with a countable transitive model. See Kunen (1980) for an exposition of this method. countable transitive models One starts with a countable transitive...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

study of Polish spaces and their Borel sets. A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. A forcing or partially...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

(also known as the Howard ordinal, or Howard–Bachmann ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

expansion is one, almost give a one-to-one mapping from subsets of a countable set (the set of positions in the expansions) to real numbers, but it fails...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum. Being an...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

the generic uncountable word “le linge” to define “laundry”, but the countable word “un/le/les linge-s” (which has no meaning in the French of France)...

February 2013). "Evolution of the cosmological horizons in a universe with countably infinitely many state equations". Journal of Cosmology and Astroparticle...

contain measurable subsets and countable unions of its elements. The concept of 𝜎-ideal is dual to that of a countably complete (𝜎-) filter. If a measure...

makes it clear the fraction must be a noun referring to a part of another countable noun. In many languages, including the East Asian languages referred to...

in a countable signature has a countable elementary substructure. Conversely, for any infinite cardinal κ every infinite structure in a countable signature...

"rice", which in fact refers to a collection of items which are logically countable. However, English has no productive process of forming singulative nouns...

fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

and is licensed by the head noun. That is, it must agree in number and countability (e.g., many people, *many person, some police, *a police) with the head...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

grammatical gender. Countable nouns inflect for number (singular and plural). However, the division between uncountable and countable nouns is more ambiguous...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

responded to these paradoxes by developing the notion of a potential countable infinity, as well as the infinitely divisible continuum. The Indian philosopher...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

infinite number of steps). Usually however, any random variable with only countably many possible values is called a discrete random variable, in this case...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

judgement of SNA statisticians. Some transactions are easily identifiable and countable, because they always occur in the same way, and for the same purpose....

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

collection of mutually disjoint non-empty open intervals in R is countable (this is the countable chain condition for the order topology of R), is R necessarily...

later) pharmacopoeiae in Late Latin form as either a countable or uncountable noun. As a countable noun, it is spelled variously as (singular) oxymellus[citation...

In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally...

4 , … . {\displaystyle n=1,2,3,4,\ldots .} That is, it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal...

scales (e.g. in hypoglycaemia and hyperthermia), as opposed to discrete or countable scales, where oligo- and poly- should be used (e.g. in oligarchy and polygamy)...

Telugu (Dravidian) Teso (Nilo-Saharan) Burushaski: masculine, feminine, countable nouns (such as animals), and uncountable nouns (which can refer to abstract...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

characteristic of being countable, by stating the following: The set R ∪ J ∪ E 2 ∪ E 3 {\displaystyle R\cup J\cup E_{2}\cup E_{3}} is countable. When I = [ a ...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

Magazine, October 1, 2011 The Hidden Girl and Other Stories 2011 The Countable Asimov's, December 2011 2011 Safe Empathy Daily Science Fiction, November...

→ B {\displaystyle X\colon A\to B} is discrete provided its image is countable. The pushforward measure X ∗ ( P ) {\displaystyle X_{*}(P)} —called the...

1143. JSTOR 71858. PMC 221287. PMID 16578557. For a proof sketch, see Countability of the Rationals Goldrei, Derek (1996). Classic Set Theory. Chapman &...

theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem...

In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. There is no standard notation for ordinals beyond...

shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of a Fourier...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

has a first element. (compare Dedekind cut) There exists a non-empty, countable subset S of C such that, if x,y ∈ C such that x < y, then there exists...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

that every open set in X is the union of sets of the class S. Second-countability is implied by the following properties (from weaker to stronger): The...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

(1969). "On the cardinality of bicompacta satisfying the first axiom of countability". Soviet Mathematics. 10: 967–970. Arkhangelskii, A. V. (1978). "Structure...

{\displaystyle \aleph _{0}} . (Analogously, the class of hereditarily countable sets is denoted by H ℵ 1 {\displaystyle H_{\aleph _{1}}} .) It can also...

of Nairobi in Kenya, he signed up for a doctorate at Edinburgh on NP countability, receiving his PhD in 1978. He arrived at Monash University in 1978,...

^{\omega }} ⁠, ϵ 0 {\displaystyle \epsilon _{0}} ..., all of which are countable ordinals. Among these, only ⁠ ω {\displaystyle \omega } ⁠ itself is an...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

used in class inheritance of object-oriented programming. Hereditarily countable set Hereditary property Hierarchy (mathematics) Nested set model for storing...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

online advertising) is when an ad is fetched from its source, and is countable. Whether the ad is clicked is not taken into account. Each time an ad...

( x ) {\displaystyle x\in f(x)} ", this is a diagonal argument. For a countable (or finite) set, the argument of the proof given above can be illustrated...

Infinite graph containing all countable graphs, the graph formed by extending the G(n, p) model to graphs with a countably infinite number of vertices....

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing...

has order p are also called extraspecial groups. The classification of countably infinite extraspecial groups is very similar to the finite case, (Newman...

density can be defined by replacing mass by number of particles or other countable quantity. Area density can be calculated as: ρ A = m A {\displaystyle...

to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense). It is used in the proof of results...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

object such as an integer, permutation or graph must be found from a countable set. A problem with continuous variables is known as a continuous optimization...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, 2 I {\displaystyle 2^{I}} is the Cantor space. All points of...

Finite number may refer to: Natural number, a countable number less than infinity, being the cardinality of a finite set Real number, such as may result...

Dodecagram - star polygon with 12 sides Apeirogon - generalized polygon with countably infinite set of sides Annulus Arbelos Circle Archimedes' twin circles...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

stitches over the printed pattern. Cross-stitch is often executed on easily countable fabric called aida cloth, whose weave creates a plainly visible grid of...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

to people, both uncountable in the sense of people as in a crowd, and countable (plural Völker) in the sense of a people as in an ethnic group or nation...

space, which means that every open cover of X {\displaystyle X} has a countable subcover. It is, therefore, enough to show that there exists some open...

computable ordinals are by definition countable. Conversely, for many countable ordinals, the "natural" witnesses for countability are also witnesses for computability...

subsets. In particular we can take A {\displaystyle A} to be a finite or countable set with all subsets being measurable. The Jensen–Shannon divergence (JSD)...

that for all positive integers n {\displaystyle n} , there is a largest countable Σ 2 n 1 {\displaystyle \Sigma _{2n}^{1}} set. Martin, Donald A.; Steel...

now called the Heine–Borel theorem. His formulation was restricted to countable covers. Pierre Cousin (1895), Lebesgue (1898) and Schoenflies (1900) generalized...

conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets and the cardinality of the set of all real numbers. The...

Finals Hard (i) RR 200 Points earned at Davis Cup 280 Points earned at Non-Countable Tournaments 45 Total year points 11010 4320 2 Overall points 6690...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

{\displaystyle a^{\omega }a=a^{\omega }} . KKM p. 29 Pin p. 158 ω-semigroup E is countable descending chain under the order a ≤H b Gril p. 233–238 Left Clifford...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

Interior Continuity Space compact connected Hausdorff metric uniform second-countable Homotopy homotopy group fundamental group Simplicial complex CW complex...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

the exact nature of which need not concern us here—it is composed of a (countable) infinity of numbers. If one knows these numbers, one can foresee the...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

000+). The ImageNet dataset filtered these to 21,841 synsets that are countable nouns that can be visually illustrated. Each synset in WordNet 3.0 has...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

"delta-ring") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti. Most introductions to probability theory...

{ZF}}} have been defined that negate the countability of such a countable union of pairs. Assuming countable choice rules out that model as an interpretation...

restricted to a finite set of symbols. A Turing machine can represent a countable set (such as the rational numbers) by strings of symbols, but this does...

\alpha } . So both the theorem and the conjecture assert that a certain countable set misses a certain set of measure zero. The theorem is not currently...

general topology, a Dowker space is a topological space that is T4 but not countably paracompact. They are named after Clifford Hugh Dowker. The non-trivial...

Determiner Countable singular nouns (e.g. notebook, week) Uncountable nouns (e.g. luggage, music) Countable plural nouns (e.g. notebooks, weeks) a / an...

refer to Equestrians ("people" uncountable) Ethnographically, ("people" countable; mostly dated, also mounted people) Eurasian nomads Plains Indians Borana...

His investigations on countable models of first-order theories led him to the Vaught conjecture stating that the number of countable models of a complete...

arbitrary new constant. The Herbrand universe is countably infinite if σ {\displaystyle \sigma } is countable and a function symbol of arity greater than 0...

of arcs (directed edges) of the graph K is a set of keys, taken from a countable set, defining the nature of attributes/properties V is a set of values...

theorem states that the cardinality of the second type class, the class of countable order types, equals the cardinality of the continuum. It was used by Felix...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

we must note, is used in two senses – both of what is counted or the countable and also of that with which we count. Time obviously is what is counted...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

works for the game. The definition generalizes from binary alphabet to countable alphabet: The frequency of each letter converges to a limit greater than...

Example of a cumulative distribution function with a countably infinite set of discontinuities...

the hyperarithmetical hierarchy is indexed by a countable ordinal number (ordinal), but not all countable ordinals correspond to a level of the hierarchy...

{\displaystyle K[x_{1},x_{2},x_{3},\dots ]} is an integral domain. It has a countably infinite set of ideal classes. If d {\displaystyle d} is a square-free...

positive integer – Representation of a number as a product of primes Countable set – Mathematical set that can be enumerated Sequence – Function of the...

avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire...

and plants, such as the number of fins or scales in fish. A meristic (countable trait) can be used to describe a particular species, or used to identify...

it (object, abstraction, or animal), and their other inflected forms. Countable and uncountable nouns are distinguished by the choice of many/much. The...

v t e Set theory Overview Set (mathematics) Axioms Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold. Given this definition,...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

first uncountable ordinal, can be constructed as follows: the set of countable well orders exists as a subset of P ( N × N ) {\displaystyle P({\mathbb...

X {\displaystyle X} along with a sigma-algebra) but is mostly used on countable sets. In formal notation, we can turn any set X {\displaystyle X} into...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

sufficient that all of the following conditions be met: G is connected. G has countable sets of vertices and edges. G has no vertices of (finite) odd degree....

realize the total vagueness of notions such as real numbers, sets, and countability, and moreover, that asking about the truth or falsity of the least upper...

Example of a cumulative distribution function with a countably infinite set of discontinuities...

≥ 2 by choosing the domain as { 0, 1/(n − 1), 2/(n − 1), ..., 1 } any countable set by choosing the domain as { p/q | 0 ≤ p ≤ q where p is a non-negative...

bounded. Every discrete space is first-countable; it is moreover second-countable if and only if it is countable. Every discrete space is totally disconnected...

guessing. The latter result is generalized by Woodward to functions on countably infinite domains. No free lunch in search and optimization No free lunch...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

space H {\displaystyle \mathbb {H} } is a fixed complex Hilbert space of countably infinite dimension (as a hilbert-basis). The observables of a quantum...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

languages, Hopi does not treat the flow of time as a sequence of distinct, countable instances, like "three days" or "five years", but rather as a single process...

correct answer than there are where 4 or fewer do so. In this variant, a countably infinite number of prisoners, each with an unknown and randomly assigned...

topology, σ-compact topological space is one that can be written as a countable union of compact subsets. In mathematical analysis and in probability...

is called weakly hereditary or closed-hereditary. For example, second countability and metrisability are hereditary properties. Sequentiality and Hausdorff...

ω-bounded space is a topological space in which the closure of every countable subset is compact. More generally, if P is some property of subspaces...

X_{1},X_{2},\dots ,X_{n}} of n {\displaystyle n} finitely (or at most countably) supported random variables on the same probability space. There are 2n...

hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices. This tiling...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

to and represent a larger amount of that thing(s). The things could be countable objects such as individual items available as units for sale, or an uncountable...

measurable set is the inf of the measures of sets containing it that are countable unions of products of measurable sets. The maximal product measure can...

curves are modular. Wiles tried at first to map elliptic curves to a countable set of modular forms. He found that this direct approach was not working...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

first-countable space then it follows that the set of i ∈ I {\displaystyle i\in I} such that a i ≠ 0 {\displaystyle a_{i}\neq 0} is countable. This need...

topological space X {\displaystyle X} is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According...

Markov chains and Markov processes in action. All examples are in the countable state space. For an overview of Markov chains in general state space,...

\mathbb {N} \vert =\aleph _{0}} ⁠ are called countable sets; these are either finite sets or countably infinite sets (sets of cardinality ⁠ ℵ 0 {\displaystyle...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

given here. In the simplest case, where the random variable X takes on countably many values (so that its distribution is discrete), the proof is particularly...

choice function. A weaker form of AC, the axiom of countable choice (ACω) states that every countable set of nonempty sets has a choice function. However...

while the exp ring is a polynomial ring in countably many generators. For each element g of G introduce a countable set of variables gi for i>0. Define exp(gt)...

into bijection with the set of all natural numbers, then it is called "countably infinite". This kind of counting differs in a fundamental way from counting...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

converse of Jensen's covering theorem is also true: if 0# exists then the countable set of all cardinals less than ℵ ω {\displaystyle \aleph _{\omega }} cannot...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter...

Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures. Carathéodory's work on outer measures found many applications...

chosen to be countable if X {\displaystyle X} is a separable space, which by definition means that X {\displaystyle X} contains some countable dense subset...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

Nominative Comitative Instrumental Classifier Measure word Construct state Countability Count noun Mass noun Collective noun Definiteness Gender Genitive construction...

v t e Large countable ordinals First infinite ordinal ω Epsilon numbers ε0 Feferman–Schütte ordinal Γ0 Ackermann ordinal θ(Ω2) small Veblen ordinal θ(Ωω)...

covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways...

principle for infinite sets: If uncountably many pigeons are stuffed into countably many pigeonholes, there will exist at least one pigeonhole having uncountably...

Use examples ê 个 The Hokkien counter word ê is available for most of countable nouns. People and Things bué/bé/bér 尾 Fish (hî/hû/hîr) châng 欉 Tree, plant...

features Noun Abstract / Concrete Adjectival Agent Animacy Bare Collective Countable Initial-stress-derived Mass Noun adjunct Proper Relational Strong / Weak...

submodule, then the ring is hereditary. A complete classification of countable reduced periodic abelian groups is given by Ulm's theorem. Several distinct...

tree contains no cycles at all, it is bipartite. Every tree with only countably many vertices is a planar graph. Every connected graph G admits a spanning...