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Topics referred to by the same term
generating set Countably generated space, a topological space in which the topology is determined by its countable subsets Countably generated module. (Kaplansky's
Countably_generated
convergent sequences. The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well
Countably_generated_space
Module generated by a countable subset
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
Countably_generated_module
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
Collapsing_algebra
Algebraic structure of set algebra
Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum). A separable measure
Σ-algebra
In algebra, module with a finite generating set
...] of all polynomials in countably many variables. R itself is a finitely generated R-module (with {1} as generating set). Consider the submodule
Finitely_generated_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 is
Kaplansky's theorem on projective modules
Kaplansky's_theorem_on_projective_modules
Group type in algebra
group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not
Finitely_generated_group
Type of probability space
\textstyle (\Omega ,{\mathcal {F}},P)} is standard if it is countably separated, countably generated, and absolutely measurable. See (Rokhlin 1952, the end
Standard_probability_space
Topological space with a dense countable subset
topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example
Separable_space
Property of topological spaces
Hausdorff. Compact-open topology – Type of topology Countably generated space Finitely generated space – Type of topology in mathematicsPages displaying
Compactly_generated_space
Generalization of mass, length, area and volume
non-increasing functions of t , {\displaystyle t,} so both of them have at most countably many discontinuities and thus they are continuous almost everywhere, relative
Measure_(mathematics)
Direct summand of a free module (mathematics)
example is R/I where R is a direct product of countably many copies of F2 and I is the direct sum of countably many copies of F2 inside of R. The R-module
Projective_module
Concept in mathematics
of a module. Countably generated module Flat module Invariant basis number "ac.commutative algebra – Existence of a minimal generating set of a module
Generating_set_of_a_module
Topological space where each point has a countable neighbourhood basis
first-countable space is compactly generated. Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space
First-countable_space
Commutative group where every element is the sum of elements from one finite subset
of cyclic groups. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. The integers
Finitely generated abelian group
Finitely_generated_abelian_group
more, the ring cannot be Jacobson. (Amitsur 1956) showed that any countably generated algebra over an uncountable field is a Jacobson ring. Tate algebras
Jacobson_ring
Ideal in a ring which has properties similar to prime elements
In a commutative ring, an ideal maximal with respect to being not countably generated is prime. Radical ideal Maximal ideal Dedekind–Kummer theorem Residue
Prime_ideal
Theory in mathematics
The set of cycles is the set of triples (H, ρ, F), where H is a countably generated graded Hilbert module over B, ρ is a *-representation of A on H as
KK-theory
continuous module countably generated A countably generated module is a module that admits a generating set whose cardinality is at most countable. cyclic A module
Glossary_of_module_theory
Function that returns cardinal numbers
t(X)=\aleph _{0}} the space X {\displaystyle X} is said to be countably generated or countably tight. The augmented tightness of a space X , {\displaystyle
Cardinal_function
Abstract algebra concept
free group in countably infinitely many generators, and so cannot be finitely generated. However, every subgroup of a finitely generated abelian group
Generating_set_of_a_group
Abstract algebra concept
Azumaya's theorem, if, in addition, each M i {\displaystyle M_{i}} is countably generated, then there is the following refinement (due originally to Crawley–Jónsson
Decomposition_of_a_module
Generalization of "n-th" to infinite cases
whose predecessors form a countably infinite set. The set of all α having countably many predecessors—that is, the set of countable ordinals—is the union
Ordinal_number
A subgroup of a group
characteristic subgroups. The lattice of verbal subgroups of the countably generated free group under inclusion is anti-isomomorphic to the lattice of
Verbal_subgroup
Mathematical objects that generalise the notion of Hilbert spaces
_{C(X)}(x):=g(\sigma (x),\rho (x)).} The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra A = C ( X
Hilbert_C*-module
Intersection of an open set and a closed set
locally closed. See Glossary of topology#S for more of this notion. Countably generated space Bourbaki 2007, Ch. 1, § 3, no. 3. Pflaum 2001, Explanation
Locally_closed_subset
Type of topological space in mathematics
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
Limit_point_compact
Boolean algebra with all operators and laws forming a complete logical system
Boolean algebra A generated by this set and then take its completion B. However B is not a "free" complete Boolean algebra generated by X (unless X is
Complete_Boolean_algebra
Topology determined by family of subspaces
the partition. Finitely generated spaces are those determined by the family of all their finite subspaces. Countably generated spaces are those determined
Coherent_topology
Technical treatment of Boolean algebras
the free Boolean algebra on countably many generators, meaning the Boolean algebra of all finitary operations on a countably infinite set of generators
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
Type of algebra
finitely generated algebra (also called an algebra of finite type) over a (commutative) ring R {\displaystyle R} , or a finitely generated R {\displaystyle
Finitely_generated_algebra
Intersection of Set Theory and General Topology
ℵ 0 {\displaystyle \aleph _{0}} the space X is said to be countably generated or countably tight. The augmented tightness of a space X, t + ( X ) {\displaystyle
Set-theoretic_topology
families of the Ind-Pro objects are countable) are equivalent to countably generated Tate R-modules in the sense of Drinfeld mentioned above. A Tate Lie
Tate_vector_space
Characterizes when a topological space is metrizable
in a space X {\displaystyle X} is countably locally finite (or 𝜎-locally finite) if it is the union of a countable family of locally finite collections
Nagata–Smirnov metrization theorem
Nagata–Smirnov_metrization_theorem
Commutative group (mathematics)
classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal
Abelian_group
Abstraction useful in the construction and triangulation of topological spaces
of the groups will be finitely generated, whereas the singular chain groups are, in general, not even countably generated. One drawback of this method is
Delta_set
First article on transfinite set theory
discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability
Cantor's first set theory article
Cantor's_first_set_theory_article
Mathematical property of a space
every sequence has a convergent subsequence. Countably compact. A space is countably compact if every countable open cover has a finite subcover. Pseudocompact
Topological_property
Class of mathematical sets
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
Borel_set
Topology made of cocountable subsets
connected, locally connected and pseudocompact, but neither weakly countably compact nor countably metacompact, hence not compact. Uncountable set: On any uncountable
Cocountable_topology
Infinite cardinal number
length, and the set of all finite subsets of any given countably infinite set. Among the countably infinite sets are certain infinite ordinals, including
Aleph_number
Infinite graph containing all countable graphs
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
Rado_graph
Type of countable group in group theory
are uncountably many finitely generated groups, and a countable group can only have countably many finitely generated subgroups. It is easy to see from
SQ-universal_group
Method in mathematical logic
the image of A in both structures). Essential countability (EC) Up to isomorphism, there are countably many structures in K {\displaystyle \mathbf {K}
Fraïssé_limit
Mathematical concept
countably additive (also called σ-additive): if { A i } i = 1 ∞ ⊆ F {\displaystyle \{A_{i}\}_{i=1}^{\infty }\subseteq {\mathcal {F}}} is a countable collection
Probability_space
Field extension that is not algebraic
separably generated if it admits a separating transcendence basis. If a field extension is finitely generated and it is also separably generated, then each
Transcendental_extension
Unary operation on string sets
{\displaystyle V} is any finite or countably infinite set of characters, then V ∗ {\displaystyle V^{*}} is a countably infinite set. As a result, each formal
Kleene_star
American mathematician Ken Dykema introduced the condition first for countably generated ideals. Uzbek and Australian mathematicians Fedor Sukochev and Dmitriy
Commutator_subspace
Broadest definition of sizes in integer-dimensional spaces
first define it on Borel sets, that is all sets that can be obtained by countably many operations of unions, set complement, and intersections, from the
Lebesgue_measure
Topology on the real numbers
real numbers (which is generated by the open intervals). The reason is that every open interval can be written as a (countably infinite) union of half-open
Lower_limit_topology
Topological space that is homeomorphic to a metric space
σ-locally finite base. A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets. For a closely related theorem
Metrizable_space
Abelian group with no non-trivial torsion elements
generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case
Torsion-free_abelian_group
Set of points on a line segment with certain topological properties
(}\subset \mathbb {N} _{0}\,3^{-\mathbb {N} _{0}}{\Bigr )}} which is a countably infinite set. As to cardinality, almost all elements of the Cantor set
Cantor_set
Certain topology in mathematics
such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements of the
Order_topology
Type of topology in mathematics
Alexandrov topology are also known as Alexandrov-discrete spaces or finitely generated spaces. The latter name stems from the fact that their topology is uniquely
Alexandrov_topology
Branch of topology
locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty. Any open subspace of a Baire space is
General_topology
Theorem in set-theoretic geometry
admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere Sn−1 can be partitioned into countably infinitely many pieces
Banach–Tarski_paradox
sets is countable. Countably compact A space is countably compact if every countable open cover has a finite subcover. Every countably compact space is
Glossary_of_general_topology
Type of measure space
is automatically countably additive. Define M {\displaystyle {\mathcal {M}}} to be the σ {\displaystyle \sigma } -algebra generated by A {\displaystyle
Loeb_space
Natural number
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
5
1 ≤ i ≤ n − 1 {\displaystyle 1\leq i\leq n-1} . An infinite path is a countably infinite sequence of edges e 1 e 2 … {\displaystyle e_{1}e_{2}\ldots }
Graph_C*-algebra
Family of subsets representing "large" sets
removed. A filter F {\displaystyle {\mathcal {F}}} is countably deep if the kernel of any countable subset of F {\displaystyle {\mathcal {F}}} belongs to
Filter_on_a_set
Theorem extending pre-measures to measures
cardinal of every non-empty set in Σ 0 {\displaystyle \Sigma _{0}} is countably infinite ( ℵ 0 {\displaystyle \aleph _{0}} ). Let μ 0 {\displaystyle \mu
Carathéodory's extension theorem
Carathéodory's_extension_theorem
Maximal proper filter
intersection of any countable collection of elements of U {\displaystyle U} is still in U {\displaystyle U} —is called countably complete or σ-complete
Ultrafilter_on_a_set
Variable representing a random phenomenon
but each y {\displaystyle y} admits at most a countable number of roots (i.e., a finite, or countably infinite, number of x i {\displaystyle x_{i}} such
Random_variable
Mathematical group that can be generated as the set of powers of a single element
finitely generated abelian group or nilpotent group is polycyclic. Cycle graph (group) Cyclic module Cyclic sieving Prüfer group (countably infinite analogue)
Cyclic_group
Directed path algebra
1 ≤ i ≤ n − 1 {\displaystyle 1\leq i\leq n-1} . An infinite path is a countably infinite sequence of edges e 1 e 2 … {\displaystyle e_{1}e_{2}\ldots }
Leavitt_path_algebra
{\displaystyle k_{i}} are integers. The finite set S generates G, so a boundedly generated group is finitely generated. An equivalent definition can be given in
Boundedly_generated_group
Descriptive set theory relation
the action. That is to say, countable Borel equivalence relations are exactly those generated by Borel actions by countable groups. This lemma is due to
Countable_Borel_relation
Probability that random variable X is less than or equal to x
function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying
Cumulative distribution function
Cumulative_distribution_function
Mathematical function for the probability a given outcome occurs in an experiment
probability distribution: for many random variables with finitely or countably infinitely many values. Probability mass function (pmf): function that
Probability_distribution
Algebraic field extension
closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite. The fundamental theorem of algebra states that
Algebraic_closure
complete theory with a finite model has no countably infinite models. The theories with just one countable model are the ω-categorical theories. There
Vaught_conjecture
Anticommutating number
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
Grassmann_number
Random process independent of past history
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
Markov_chain
Family closed under subsets and countable unions
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
Sigma-ideal
Locally compact topological group with an invariant averaging operation
all its finitely generated subgroups are. That is, locally amenable groups are amenable. By the fundamental theorem of finitely generated abelian groups
Amenable_group
Binary sequence
null cover covers a measure 0 set, there are only countably many constructive null covers, and a countable union of measure 0 sets has measure 0. This implies
Algorithmically random sequence
Algorithmically_random_sequence
Special case in probability theory; introduces tail events
of countably infinite families of σ-algebras. For illustrative purposes, we present here the special case in which each sigma algebra is generated by
Kolmogorov's_zero–one_law
every finitely generated subgroup of a finitely presented group is recursively presented, so the recursively presented finitely generated groups are (up
Higman's_embedding_theorem
Collection of open sets used to define a topology
generated by Γ {\displaystyle \Gamma } , which is the Euclidean topology on R {\displaystyle \mathbb {R} } , is coarser than the topology generated by
Base_(topology)
Specification of a mathematical group by generators and relations
this we can deduce that there are (up to isomorphism) only countably many finitely generated recursively presented groups. Bernhard Neumann has shown that
Presentation_of_a_group
Order-preserving mathematical function
jump and removable discontinuities. f {\displaystyle f} can only have countably many discontinuities in its domain. The discontinuities, however, do not
Monotonic_function
Property in general topology
the FIP; this family is called the principal filter on X {\textstyle X} generated by K {\textstyle K} . The subset B = { I ⊆ R : K ⊆ I and I an open
Finite_intersection_property
Average value of a random variable
With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider the distinct
Expected_value
Two-dimensional manifold
the Cantor set. M may have a finite or countably infinite number Nh of handles, as well as a finite or countably infinite number Np of projective planes
Surface_(topology)
Group theory
{\displaystyle \mathbb {Z} } generates a countable simple group which has the following properties: It is not finitely generated. Every finite group, every
Residue-class-wise affine group
Residue-class-wise_affine_group
Type of mathematical space
equivalent to compactness for first-countable uniform spaces). (X, d) is limit point compact (also called weakly countably compact); that is, every infinite
Compact_space
Concept in topology
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
Polish_space
words, the σ–algebra of Baire sets is the σ–algebra generated by all those intersections of countably many open sets that yield a compact set. Alternatively
Baire_set
Operations on ordinals that extend classical arithmetic
relation on the ordinals greater than 1, and all equivalence classes are countably infinite. Distributivity holds, on the left: α ⋅ (β + γ) = α ⋅ β + α ⋅
Ordinal_arithmetic
finitely-generated group with polynomial growth on a standard Borel space induces a hyperfinite orbit equivalence relation. The action of a finitely-generated
Hyperfinite equivalence relation
Hyperfinite_equivalence_relation
X {\displaystyle X} be a compactly generated Hausdorff space (e.g., a locally compact space or a first countable Hausdorff space), B {\displaystyle {\mathcal
Filters_in_topology
Eighteenth letter of the Greek alphabet
to represent unknown angles, additionally serving as a shorthand for "countably", whereas Σ is regularly used as the operator for summation, e.g.: ∑ k
Sigma
Mathematician (1845–1918)
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
Georg_Cantor
Family of sets closed under intersection
𝜎-algebra generated by that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they hold for the generated 𝜎-algebra
Pi-system
Categories of symbols in formal grammars
language that contains countably infinite many finite-length words by the fact that we can apply the first rule any countable times as we wish. Diagram
Terminal and nonterminal symbols
Terminal_and_nonterminal_symbols
Isometry group of Euclidean space
group generated by a translation of 1 and one of √2, and, in 2D, the group generated by a rotation about the origin by 1 radian. Non-countable groups
Euclidean_group
Branch of mathematical logic
union of a perfect set and a countable set). Silver's dichotomy (every coanalytic equivalence relation has either countably many equivalence classes or
Reverse_mathematics
COUNTABLY GENERATED
COUNTABLY GENERATED
Surname or Lastname
English and Irish
English and Irish : from the Old Norse personal name þorkell, a contracted form of a name composed of the elements þórr, name of the Scandinavian god of thunder (see Thor) + ketill ‘cauldron’. The personal name Thurkill or Thirkill was in use throughout England in the Middle Ages; in northern England it had been introduced directly by Scandinavian settlers, whereas in the South it was the result of Norman influence. This surname and its variants are especially common in East Anglia. In Ireland the Old Norse name was adopted as a Gaelic personal name (Thorcall), which generated the surnames McCorkle and Corkill.
Boy/Male
Hindu, Indian
Self Generated; Lord Shiva
Boy/Male
Indian, Telugu
Generated
Girl/Female
French
From the countly estate.
Girl/Female
French
From the countly estate.
Boy/Male
Hindu, Indian
Un Countable; Multiple; Countless
COUNTABLY GENERATED
COUNTABLY GENERATED
Surname or Lastname
English
English : perhaps a reduced form of Hardwick.
Boy/Male
Hindu, Indian
Pure Joy
Boy/Male
Hindu, Indian, Tamil
King
Girl/Female
Hindu, Indian, Traditional
Like a Goddess
Boy/Male
Hindu, Indian, Marathi, Punjabi, Sikh
Raised by God
Boy/Male
Muslim
Standing tall like a mountain, Ability to withstand all that is thrashed upon it
Girl/Female
Indian, Punjabi, Sikh
Intimate Love
Surname or Lastname
English
English : from the Middle English personal name Alstan, which is a coalescence of several different Old English personal names: Æ{dh}elstÄn ‘noble stone’, ÆlfstÄn ‘elf stone’, EaldstÄn ‘old stone’, or EalhstÄn ‘altar stone’.English : habitational name from any of various places called Alston (in Cumbria, Lancashire, Devon, and Somerset) or Alstone (in Gloucestershire and Staffordshire). With the exception of Alston in Cumbria, which is formed with the Old Scandinavian personal name Halfdan, these place names all consist of an Old English personal name + Old English tÅ«n ‘settlement’, for example Ælfsige in the case of Alstone in Gloucestershire.English : In 1682 John Alston of Hammersmith, Middlesex, England, began a seven-year apprenticeship to James Jones, merchant, of Charleston, SC. He had many prominent descendants, among whom the name is often spelled Allston.
Girl/Female
Gaelic
Joy.
Boy/Male
Hindu, Indian, Marathi, Parsi, Sanskrit, Thai
Wisest Amongst the Wise
COUNTABLY GENERATED
COUNTABLY GENERATED
COUNTABLY GENERATED
COUNTABLY GENERATED
COUNTABLY GENERATED
adv.
In an accountable manner.
n.
The part of a spiral generated in one revolution of the straight line about the pole. See Spiral, n.
n.
A curve in the form of the figure 8, with both parts symmetrical, generated by the point in which a tangent to an equilateral hyperbola meets the perpendicular on it drawn from the center.
n.
The state or quality of being numerable or countable.
a.
First born, made, or generated; original; primary; elemental; as, primogenial light.
a.
Such as can be mounted.
n.
A body or figure approaching to a sphere, but not perfectly spherical; esp., a solid generated by the revolution of an ellipse about one of its axes.
a.
Generated by water.
n.
The surface of constant negative curvature generated by the revolution of a tractrix. This surface corresponds in non-Euclidian space to the sphere in ordinary space. An important property of the surface is that any figure drawn upon it can be displaced in any way without tearing it or altering in size any of its elements.
a.
Capable of being numbered.
n.
Air or gas generated in the stomach or bowels; flatulence; as, to be troubled with wind.
v. t.
A system of compromises in the tuning of organs, pianofortes, and the like, whereby the tones generated with the vibrations of a ground tone are mutually modified and in part canceled, until their number reduced to the actual practicable scale of twelve tones to the octave. This scale, although in so far artificial, is yet closely suggestive of its origin in nature, and this system of tuning, although not mathematically true, yet satisfies the ear, while it has the convenience that the same twelve fixed tones answer for every key or scale, C/ becoming identical with D/, and so on.
n.
The place where anything is generated or produced.
a.
First formed or generated; original; primigenial.
n.
A warped surface which may be generated by a straight line moving in such a manner that every point of the line shall have a uniform motion in the direction of another fixed straight line, and at the same time a uniform angular motion about it.
n.
A hot, dry, suffocating, dust-laden wind, that blows occasionally in Arabia, Syria, and neighboring countries, generated by the extreme heat of the parched deserts or sandy plains.
n.
A solid generated by the revolution of a curved line about its base or double ordinate or chord.
a.
Not begot; not yet generated; also, having never been generated; self-existent; eternal.
a.
Capable of being produced, brought forward, brought forth, generated, made, or extended.
n.
The solid generated by the revolution of a circle, or other figure, about an exterior straight line (as an axis) lying in the same plane as the circle or other figure.