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is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable if...

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

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

topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly...

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:...

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...

a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X {\displaystyle...

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...

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...

mathematics, a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence ( x n ) n = 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} ...

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

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

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

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...

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

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...

(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...

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...

and philosophy, Skolem's paradox is the apparent contradiction that a countable model of first-order set theory could contain an uncountable set. The...

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...

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

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...

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...

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

bound) of all countable ordinals. In the von Neumann representation, the elements of ω 1 {\displaystyle \omega _{1}} are the countable ordinals (including...

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}...

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...

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...

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

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

+ 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...

_{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...

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

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...

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...

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

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...

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...

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

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

that: The set T is uncountable. The proof starts by assuming that T is countable. Then all its elements can be written in an enumeration s1, s2, ... ,...

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...

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...

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

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...

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...

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...

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...

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

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

In particular, many authors define them to be paracompact or second-countable. In the remainder of this article a manifold will mean a topological manifold...

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...

discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability...

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...

[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...

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...

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 )...

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...

W=\bigcup _{n<\omega }\{w\in W:|w|<n\}} would be a countable union of countable sets, thus countable). If p ( e 1 ) = b 1 {\displaystyle p(e_{1})=b_{1}}...

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...

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...

using the accusative case instead, assumes that there was a specific countable portion of water that was completely drunk. The work of logicians like...

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...

of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps...

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...

Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal...

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...

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...

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...

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...

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

it is possible to show there are different sizes of infinity. A set is countably infinite if it can be placed in one-to-one correspondence with the set...

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...

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...

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...

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

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

In general topology, an Fσ set (pronounced F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French:...

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

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

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

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...

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

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...

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...

Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used...

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:...

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

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...

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...

the literature. For instance, an example of a first-countable space which is not second-countable is counterexample #3, the discrete topology on an uncountable...

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...

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...

sets is called a 𝜎-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation. Let R {\displaystyle {\mathcal {R}}}...

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...

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)...

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...

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

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

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...

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...

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

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

is a countable set of dense subsets of P then there exists a D-generic filter F in P such that p ∈ F. Let p ∈ P be given. Since D is countable, D = { Di |...

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...

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...

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...

this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is...

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...

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

Now ℵ 1 {\displaystyle \aleph _{1}} is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating...

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...

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

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

and complex numbers are transcendental, since the algebraic numbers are countable, while the real numbers ⁠ R {\displaystyle \mathbb {R} } ⁠ and complex...

sets is countable. Countably compact A space is countably compact if every countable open cover has a finite subcover. Every countably compact space is...

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

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...

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...

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...

set first-countable space: every point has a countable neighbourhood basis (local base) second-countable space: the topology has a countable base separable...

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...

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...

simplicity and intuitiveness. The language's alphabet consists of: A countably infinite number of variables used for representing sets The logical connectives...

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...

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

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...

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

Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included...

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

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

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

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

purpose, may be finite (e.g., the alphabet of letters "a" through "z"), countable (e.g., { v 1 , v 2 , … } {\displaystyle \{v_{1},v_{2},\ldots \}} ), or...

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...

(monotone) function are necessarily jump discontinuities and there are at most countably many of them. Usually, this theorem appears in literature without a name...

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...

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

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

that there is some countably infinite subset of the real numbers, whose members we will call internal numbers, and some countably infinite collection...

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...

subsets of X {\displaystyle X} whose complements in X {\displaystyle X} are countable. Cofinite topology Double-pointed cofinite topology Ordinal number topology...

(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...

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

is uncountably infinite but the set of all real algebraic numbers is countably infinite. His proof does not use his famous diagonal argument, which he...

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

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

point of Fraïssé's construction is to show how one can approximate a (countable) structure by its finitely generated substructures. Given a class K {\displaystyle...

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...

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

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...

hand there are countable transitive models of ZFC' (this is provable in ZFC), and the set of real numbers in such a model will be a countable set. The paradox...

_{0}} is still countable, being a countable union of countable ordinals; in fact, ε β {\displaystyle \varepsilon _{\beta }} is countable if and only if...

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

multiplication operations of a countable nonstandard model of PA. There is only one possible order type of a countable nonstandard model. Letting ω be...

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...

In mathematics, the Feferman–Schütte ordinal (Γ0) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such...

"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...

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

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...

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

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

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

numbers is the same size (cardinality) as N, and is said to be countable. Examples of countably infinite sets are the natural numbers, the even numbers, the...

theorem Well-ordering theorem Zorn's lemma Axiom of global choice Axiom of countable choice Axiom of dependent choice Boolean prime ideal theorem Axiom of...

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

is the same size (cardinality) as the set of integers: they are both countable sets. Cantor gave two proofs that the cardinality of the set of integers...

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...

Urysohn's metrization theorem, second-countable then implies metrizable. Conversely, a compact metric space is second-countable. There are many natural examples...

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...

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...

not algebraic are called transcendental and include π and e. There are countably infinite algebraic numbers, hence almost all real (or complex) numbers...

\infty }F_{n}=\Gamma {\text{-}}\limsup _{n\to \infty }F_{n}=F} . In first-countable spaces, the above definition can be characterized in terms of sequential...

{\displaystyle B\setminus A\in D;} D {\displaystyle D} is closed under countable increasing unions: if A 1 ⊆ A 2 ⊆ A 3 ⊆ ⋯ {\displaystyle A_{1}\subseteq...

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...

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

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...

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

implicitly, that as a topological space a surface is also nonempty, second-countable, and Hausdorff. It is also often assumed that the surfaces under consideration...

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...

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

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...

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

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

the following statement: MA(κ) For any partial order P satisfying the countable chain condition (hereafter ccc) and any set D = {Di}i∈I of dense subsets...

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...

choice to select one such at each step. For inductions and recursions of countable length, the weaker axiom of dependent choice is sufficient. Because there...

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...

abbreviated as V=L Axiom of countability Every set is hereditarily countable Axiom of countable choice The product of a countable number of non-empty sets...

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...

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 ...

_{i\in I}{\overline {A_{i}}},\end{aligned}}} where I is some, possibly countably or uncountably infinite, indexing set. In set notation, De Morgan's laws...

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

cocountable subsets of X {\displaystyle X} (those whose complement is countable) form a filter, the cocountable filter which is finer than the Fréchet...

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...

This then opens the question as to what sort of function from a countable set to a countable set, such as f and g above, can actually be constructed. Different...

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

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...

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...

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...

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

measure-preserving actions of countable amenable groups on probability spaces. The infinite tensor product of a countable number of factors of type In...

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

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

example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models. ZFC, NBG, and MK each have models...

More generally, an assignment of elements from a countable mathematical object, such as a countable group, to natural numbers to allow algorithmic manipulation...

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...

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...

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...

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...

the set of natural numbers {0, 1, ...}, and α ranges up to some large countable ordinal). A primary example is the Wainer hierarchy, or Löb–Wainer hierarchy...

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...

has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not...

M {\displaystyle M} of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means M {\displaystyle...

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

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

which characterises classes of finite structures that arise as ages of countable homogeneous structures. The diagram of the amalgamation property appears...

function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger...

a real vector space whose intersection with every measure-zero set is countable. The existence of Sierpiński sets is independent of the axioms of ZFC...

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...

said to be countably infinite. Likewise, any infinite set that has a bijection with the integers or the rational numbers is also countably infinite, since...

( 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...

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

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

index set) has a convergent subsequence if and only if there exists a countable set K ⊆ I {\displaystyle K\subseteq I} such that ( x m ) m ∈ K {\displaystyle...

example, in the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} take a countably infinite discrete set A in the unit disk, with the points getting denser...

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Matroid Greedoid Ramsey theory Van der Waerden's...

\times \mathbb {R} \times \cdots } can be visualized as a vector with countably infinite real number components. This set is frequently denoted R ω {\displaystyle...

(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...

linearly-ordered Heyting algebras logics of (classes of) linearly ordered and countable intuitionistic Kripke structures with constant domains logics of relative...

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

{\displaystyle X} such that X × Y {\displaystyle X\times Y} is countably compact for every countably compact space Y {\displaystyle Y} . Frolík prepared his...

countable too, and note that the successor set of any countable set is countable. The set ω {\displaystyle \omega } is trivially infinite, countable and...

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

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

the finite particular point topology. If X is countably infinite, the topology on X is called the countable particular point topology. If X is uncountable...

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

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

diamond principle. One states that there is a countable collection Aα of subsets of α for each countable ordinal α such that for any subset A of ω1 there...

𝜎-ideal – Family closed under subsets and countable unions 𝜎-ring – Family of sets closed under countable unions The nullary (0-ary) intersection of...

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

\operatorname {diam} U_{i}<\delta \right\},} where the infimum is taken over all countable covers U {\displaystyle U} of S {\displaystyle S} . The Hausdorff d-dimensional...

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...

according to the properties they have. Empty set Finite set, Infinite set Countable set, Uncountable set Power set Closed set Open set Clopen set Fσ set Gδ...

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

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...

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

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...

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

be stable or unstable; be ω-stable (same as totally transcendental for countable theories); be superstable have an atomic model; have a prime model; have...

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)...

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

+ 1 {\displaystyle K_{i+1}} , meaning the space is σ-compact (i.e., a countable union of compact subsets.) If there is an exhaustion by compact sets,...

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,...

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

\left(X,{\mathcal {F}}_{2}\right).} If X {\displaystyle X} is finite or countably infinite, the σ {\displaystyle \sigma } -algebra is most often the power...

example, a set that contains only itself is a hereditary set. Hereditarily countable set Hereditarily finite set Well-founded set Kunen, Kenneth (1980). Set...

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

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...

example is 0′, the degree of the halting problem. Every Turing degree is countably infinite, that is, it contains exactly ℵ 0 {\displaystyle \aleph _{0}}...

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...

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

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

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

{\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...

the length of the interval In.) Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every...

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

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...

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}...

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

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

ψ0(Ωω), widely known as Buchholz's ordinal[citation needed], is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical...

conversely if p is not isolated then there is a countable model omitting p (provided that the language is countable). Example: In the theory of algebraically...

Children 3 Scientific career Fields Mathematics Thesis K-Analytic Spaces and Countable Operations in Topology (1970) Doctoral advisor Donald A. Dawson...

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

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...

A_{2}\cap A_{3}\cap \cdots } ". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on...

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

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

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

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

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...

{\displaystyle i<j} so that T i ≤ T j {\displaystyle T_{i}\leq T_{j}} . For a countable label set X {\displaystyle X} , Kruskal's tree theorem can be expressed...

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

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...

S ) = lim sup δ ↓ 0 { ∑ i ∈ I d i a m ( B i ) s | { B i } i ∈ I  is a countable collection of pairwise disjoint closed balls with diameters  ≤ δ  and...

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...

topological complexity of the type spaces. However, Morley showed that (for countable theories) this topological restriction is equivalent to a cardinality...

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...

proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi function...

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...

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...

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

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

providing one-to-one mappings in both directions between subsets of a countably infinite set and real numbers, and applying the Cantor–Bernstein–Schroeder...

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

In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal...

step n is performed at 2−n minutes before noon. This guarantees that a countably infinite number of steps is performed by noon. Since each subsequent step...

serial dilutions of samples are plated to ensure that they will yield a countable number of colonies. Plating volumes generally range from 100 μL to 1 mL...

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)...

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

represent an approximation of a real number. Because the real numbers are not countable, computers cannot represent them exactly using a finite amount of information...

automorphism (i.e. essentially up to the notation of its elements). F is countable and contains a single copy of each of the finite fields GF(2n); the copy...

ambiguity in the actual cardinality. In the case of countable languages, all prime models are at most countably infinite. There is a duality between the definitions...

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

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

strings; this form of notation is called an "axiom schema" (i.e., there is a countable number of specific forms the notation could take). This can be read in...

generalized to infinite graphs in which every vertex has finite degree. If a countably infinite graph G has no odd-degree vertices, then it may be written as...

Moreover, it is possible to consider a simpler form of saturation called countable saturation. This simplified approach is also more suitable for use by...

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

the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using...

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

\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...

In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative...

, ...), truth values (T or F), etc. A set of individual variables: A countably infinite amount of symbols representing variables used for representing...

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)...

whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove...

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...

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

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

sequence. The axiom of dependent choice says that we can form a whole (countably infinite) sequence this way. The axiom D C {\displaystyle {\mathsf {DC}}}...

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

satisfied: Any/both of the following equivalent conditions hold: Closed under countable unions: ⋃ i = 1 ∞ F i := F 1 ∪ F 2 ∪ ⋯ ∈ F {\displaystyle \bigcup _{i=1}^{\infty...

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

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

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...

E} is of Hausdorff dimension m {\displaystyle m} , and there exist a countable collection { f i } {\displaystyle \{f_{i}\}} of continuously differentiable...

other motivating subjects. Let Γ be a second countable locally compact group (for instance a countable discrete group). One can define a morphism μ :...

model described by the theory; this model will be countable if the language of the theory is countable. However this "explicit construction" is not algorithmic...

Cartesian powers of one of these spaces. A similar extension is possible for countable powers and to products of powers of Cantor space and powers of Baire space...

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

measure of 0 {\displaystyle 0} to the set of rational numbers because it is countable. Any set which has a well-defined Lebesgue measure is said to be "measurable"...

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

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...

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...

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...

theorem does not hold almost surely if the random variable has an infinite countable probability space; however, this depends on allowing a very broad range...

respect to the Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometrically, a discrete...

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...

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...

elementary embedding M is countable and transitive π(λ) = κ σ : M → N is an elementary embedding with critical point λ N is countable and transitive ρ = M...

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...

finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is an injective function mapping the...

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

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

groups. While countable torsion abelian groups are well understood through simple presentations and Ulm invariants, the case of countable mixed groups...

\{\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...

x\preceq y} and x ⪰ y {\displaystyle x\succeq y} ) are a countable set. 2. There is a countable subset of X, Z = { z 0 , z 1 , . . . } {\displaystyle Z=\{z_{0}...

plane is first countable, but not second countable or Lindelöf. The Moore plane is not locally compact. The Moore plane is countably metacompact but...

Stammbach, he proved there exists a non-free parafree group with every countable subgroup being free. Baumslag, Gilbert, Groups with the same lower central...

\ \omega ^{\omega ^{\omega }},\ \ldots } It is a countable ordinal much smaller than large countable ordinals. To express ordinals in the language of...

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

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...

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

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

of 0, while Q ∩ [ 0 , 1 ] {\displaystyle \mathbb {Q} \cap [0,1]} , a countable union of them, is not Jordan-measurable. For this reason, some authors...

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...

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

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

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...

surjection from S {\displaystyle S} to the computable numbers. There are only countably many Turing machines, showing that the computable numbers are subcountable...

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

denote the set of all functions from [ 0 , 1 ] {\displaystyle [0,1]} to countable subsets of [ 0 , 1 ] {\displaystyle [0,1]} . (In other words, A = ( [...

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

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

evaluating the overall costs of advertising campaigns. For media without countable views, CPM reflects the cost per 1000 estimated views of the ad. This...

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...

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...

is not needed; as described below, the axiom of countable choice suffices. If the graph is countable, the vertices are well-ordered and one can canonically...

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

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...

numbers is uncountably infinite but the set of all algebraic numbers is countably infinite, so there is an uncountably infinite number of transcendental...

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

theorem). Embedding between countable scattered linear order types is a well-quasi-order (Laver's theorem). Embedding between countable boolean algebras is a...

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...

works in 1934. Pontryagin's treatment relied on the groups being second-countable and either compact or discrete. This was improved to cover the general...

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

Marketplace with Kai Ryssdal, and then Managing Editor of the civic-tech app, Countable. Seabrook has since moved to Costa Rica and founded a bilingual newspaper...

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

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

can belong both to the countable and to the uncountable class, producing differences in meaning. For example, when countable, báalt means 'apple' but...

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...

functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between...

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

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...

B_{1}} is the open ball of radius 1 centered at the origin. Every second-countable manifold admits an adequate atlas. Moreover, if V = ( V j ) j ∈ J {\displaystyle...

R 3 {\displaystyle {\mathbb {R} ^{3}}} ). It states that if a certain countable subset is removed from S 2 {\displaystyle {S^{2}}} , then the remainder...

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

μ 0 ( ∅ ) = 0 {\displaystyle \mu _{0}(\varnothing )=0} and, for every countable (or finite) sequence A 1 , A 2 , … ∈ R {\displaystyle A_{1},A_{2},\ldots...

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

process is R × S where X(t) takes real values and J(t) takes values in some countable set S. For the case where J(t) takes a more general state space the evolution...

with probability at least ε {\displaystyle \varepsilon } . Let Ω be a countable state space. The kernel K is defined by the one-step conditional transition...

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

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...

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

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...

also refer to: Cantor–Bernstein theorem: cardinality of the class of countable order types equals the cardinality of the continuum Cantor–Bernstein–Schröder...

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

' 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...

information in a stochastic process. For stochastic processes with a countable index, the entropy rate H ( X ) {\displaystyle H(X)} is the limit of the...

although introduced here as an outer measure (only countably subadditive), becomes a full measure (countably additive) if restricted to the Borel sets. Evans...

} is homeomorphic to the one-point compactification of the union of a countable family of disjoint open intervals. The Hawaiian earring is a one-dimensional...

minimum? Vaught conjecture: the number of countable models of a first-order complete theory in a countable language is either finite, ℵ 0 {\displaystyle...

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...

(topological) manifold is a second countable Hausdorff space that is locally homeomorphic to a Euclidean space. Second countable and Hausdorff are point-set...

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

empty set, true arithmetic also has 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} countable models. Since the theory is complete, all of its models are elementarily...

type, because even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them...

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...

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...

subgroups of all countable ranks. Any group that acts on a tree, freely and preserving the orientation, is a free group of countable rank (given by 1...

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

is a Cantor space. But the canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space {0, 1}. This...

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...

\mathrm {rad} (B)} denotes the radius of the ball B. Then there exists a countable sub-collection G ⊂ F {\displaystyle \mathbf {G} \subset \mathbf {F} }...

depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible quantity"; that...

finite number of generators, typically n = 1, 2, 3 or 4, and those with a countably-infinite number of generators. These two situations are not as unrelated...

complete theory with less than continuum many nonisomorphic countable models can have only countably many. Vaught's conjecture, named after Robert Lawson Vaught...

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

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

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...

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...

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...

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...

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

This means, in particular, the set of tautologies over a fixed finite or countable alphabet is a decidable set. As an efficient procedure, however, truth...

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

Mexican charter airline (ICAO code: SVO) Small Veblen ordinal, a large countable ordinal Social value orientations, a psychological construct Sparse voxel...

Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure in at most countable signature; the upward Löwenheim–Skolem...

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...

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

Infinite-dimensional Lebesgue measures do not exist. If X and Y are second-countable, Hausdorff topological spaces, then the set of Borel subsets B ( X × Y...

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...

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

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

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

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...

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

Euclidean spaces by taking a countable set of coordinate patches. The conclusion of the theorem is a local statement, since a countable union of sets of measure...

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...

under countable unions. Instead the algebra A {\displaystyle {\mathcal {A}}} has the property that if a set in it is the union of a countable family...

Hausdorff space and Lebesgue measure on [a, b] is replaced by a finite countably additive measure μ on the Borel algebra of X whose support is X. This...

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

except for a very limited number of examples. Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case...

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

function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e., f ( t ) = lim n → ∞ f n ( t )  for almost every ...

"diagonalization" argument establishes the sequential compactness of a countable product of sequentially compact spaces. However, the product of continuum...

diagonal argument, used to prove that the set of real numbers is not countable Diagonal lemma, used to create self-referential sentences in formal logic...

{\displaystyle (X,\Sigma ,\mu )} be a Radon measure space and Y be a second-countable topological space equipped with a Borel algebra, and let f : X → Y {\displaystyle...

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

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...

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

variables, corresponding to whether a random variable is valued in a countable subset or in an interval of real numbers. There are other important possibilities...

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...

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

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

{\mathcal {F}}{K}} , which is usually a non-countable set, whereas compact operators have discrete countable spectra). An inhomogeneous Fredholm equation...

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...

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...

quantities, the particle number is a dimensionless quantity, specifically a countable quantity. It is an extensive property, as it is directly proportional...

Define E = S ∩ (a,b). Then E is an open set, and it may be written as a countable union of disjoint intervals E = ⋃ k ( a k , b k ) {\displaystyle E=\bigcup...

Löwenheim–Skolem theorem, which states that if a countable first-order theory has an infinite model, then it has a countable model. His 1920 proof employed the axiom...

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

distinct segments, or strata. Though this feature most often distinguishes countability in nouns and aspect in verbs, it applies more generally to any syntactic...

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

measure-theoretic sense, a σ-ring is a ring closed under countable unions, and a δ-ring is a ring closed under countable intersections. Explicitly, a σ-ring over X...

painted With the teachings of my parents Although the stars in the sky Are countable The teachings of my parents Are not Just as ships that run in the night...

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

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...

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...

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

I:a_{i}>0\right\}=\bigcup _{n=1}^{\infty }A_{n}} is countable. If I {\displaystyle I} is countably infinite and enumerated as I = { i 0 , i 1 , … } {\displaystyle...

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

they did make quantitative predictions from the premises that rays were countable objects, which were conserved in their interactions with matter (except...

Vaught (1963, p. 309), states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1,ω). A model is of type...

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...

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

C*-algebras with lattice-ordered dimension group and all isomorphism classes of countable MV algebras. Some instances of this correspondence include: There are...

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...

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

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...