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...

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

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

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

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:...

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

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...

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 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...

(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...

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

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...

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

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

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}...

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

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...

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

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...

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

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...

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...

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...

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...

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...

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...

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

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

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

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...

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...

+ 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...

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

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...

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...

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...

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...

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...

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...

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

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...

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...

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 )...

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...

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

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...

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...

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

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...

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, ... ,...

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...

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}}...

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...

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...

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...

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...

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

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

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...

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

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...

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...

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...

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...

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

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

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

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...

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

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

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...

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...

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...

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...

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...

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...

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...

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

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...

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)...

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...

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

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:...

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

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...

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

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

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...

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:...

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...

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...

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...

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

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

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

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

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

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 |...

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

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...

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...

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...

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

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...

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...

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...

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

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

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

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...

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...

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...

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

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...

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...

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...

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

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

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...

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

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...

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...

"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...

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

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...

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

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...

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...

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

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

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

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

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

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...

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

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...

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

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

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...

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...

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...

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...

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...

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

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...

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

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

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

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...

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...

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...

_{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...

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...

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

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...

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

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...

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

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...

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

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

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

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...

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...

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...

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

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...

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

(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...

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...

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...

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

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...

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

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

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

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

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

(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...

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...

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

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

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...

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

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

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...

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

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...

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

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

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

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

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...

\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...

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

{\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...

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

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)...

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...

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...

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

\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...

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...

+ 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,...

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...

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

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...

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

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...

second-order sort], there exists a countable β-model M such that X ∈ M {\displaystyle X\in M} .p. 253 (Countable ω-models are represented by their sets...

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...

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

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...

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...

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

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}}}...

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...

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

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

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...

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 ...

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...

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...

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...

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...

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

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...

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

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...

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...

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

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

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...

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...

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...

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...

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

\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...

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...

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

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...

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...

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...

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

{\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...

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...

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

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

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

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

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

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...

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...

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...

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...

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...

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

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

and complement being −x−1. There are only countably many integers, so this infinite Boolean algebra is countable. The atoms are the powers of two, namely...

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...

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

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

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

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

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...

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

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

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...

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

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...

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

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

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

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...

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...

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

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

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

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...

\{\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...

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

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

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

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...

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}...

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...

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...

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

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

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...

{\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...

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

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...

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

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

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

sequences with componentwise addition, that is, the direct product of countably infinitely many copies of Z {\displaystyle \mathbb {Z} } . It can equivalently...

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...

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...

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

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...

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...

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

' 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...

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...

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,...

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)...

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

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

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...

\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...

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...

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...

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...

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

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)...

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)...

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

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

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...

≥ 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...

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...

module over an arbitrary ring is a direct sum of countably generated projective modules. Show that a countably generated projective module over a local ring...

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

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

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

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...

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...

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...

that is at most countable has Lebesgue measure zero, and thus a bounded function (on a compact interval) with only finitely or countably many discontinuities...

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...

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...

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

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...

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}}...

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

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...

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

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...

In algebra, Hall's universal group is a countable locally finite group, say U, which is uniquely characterized by the following properties. Every finite...

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

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...

{\displaystyle X} is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then x ∈ X {\displaystyle x\in X} is a limit point of S {\displaystyle...

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...

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...

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}...

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...

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...

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...

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

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...

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...

around x {\displaystyle x} with radius 1 / n {\displaystyle 1/n} form a countable neighbourhood basis B = { B 1 / n : n = 1 , 2 , 3 , … } {\displaystyle...

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

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

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...

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

other sets (defining, for example, ℵ 0 {\displaystyle \aleph _{0}} for countable sets, ℵ 1 {\displaystyle \aleph _{1}} and so on for larger infinities)...

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...

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

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...

is a tree of height ω1 such that every branch and every antichain is countable. They are named after Mikhail Yakovlevich Suslin. Every Suslin tree is...

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

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...

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

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...

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

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

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

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...

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

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...

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...

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...

cluster graphs also form one of only a small number of different types of countably infinite homogeneous graphs. A subcoloring of a graph is a partition of...

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

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

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...

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...

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...

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...

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...

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

able to accurately capture thoughts to the difference in size between countable and uncountable sets. Cory M. Hudson characterized "Good Old Neon" as...

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...

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...

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...

University in 2001. His doctoral dissertation, Thermodynamic Formalism for Countable Markov Shifts, was supervised by Jon Aaronson. From 2000 to 2003, he was...

have property (T). If G has property (T) then G/[G, G] is compact. Any countable discrete group with property (T) is finitely generated. An amenable group...

among the reals, whereas the interval (0, 1) is not nowhere dense. A countable union of nowhere dense sets is called a meagre set. Meagre sets play an...

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...

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...

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

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

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...

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

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...

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...

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)...

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

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...

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

countable union of open intervals. Lindelöf's lemma is also known as the statement that every open cover in a second-countable space has a countable subcover...

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...

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

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...

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...

preference-relation is called: Countable - if the set of equivalence classes of the indifference relation ≃ {\displaystyle \simeq } is countable. Separable - if there...

Alexandrov-discrete spaces is Alexandrov-discrete. Every Alexandrov topology is first countable (since every point has a smallest neighborhood). Every Alexandrov topology...

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...

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

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

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...

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...

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...

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

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...

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...

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...

pattern theory, a posterior probability model was constructed over the countable union of sample space; this is therefore a hybrid system model, containing...

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

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...

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

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

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

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...

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

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...

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

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...

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...

the singular cardinals hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals. In particular, then...

(a countable intersection of dense open sets), with the dual concept being a closed nowhere dense set, or more generally a meagre set (a countable union...

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)...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

{\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } with at most countable discontinuities the set of finite subsets of real numbers the set of all...

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...

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...

summable family in X , {\displaystyle X,} then necessarily all but a countable collection of x α {\displaystyle x_{\alpha }} 's are 0. Absolutely summable...

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

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

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...

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

2015). "NCAA Updates "Countable Opponents"". The Crunch Zone. Retrieved January 1, 2026. "Evaluations of Countable & Non-Countable Opponents". NAIA. Retrieved...

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

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

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

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

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

the bonding maps, then its inverse limit is a continuum. A finite or countable product of continua is a continuum. Linear continuum Menger sponge Shape...

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...

complexity, and showing (under certain assumptions) that there are largest countable sets of certain complexities. Given a pointset A contained in some product...

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