Mathematical logic and foundations Definition
Family of sets
Builds on Set
Definition
Let be a set. A family of sets indexed by assigns to each a set . It is written . The set is the index set.
Remarks
Formally, a family indexed by is a function with domain . The same set can appear for two different indices: with is allowed. Example: for is a family indexed by . Some authors use "family of sets" for any set of sets, without an index.
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To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Element definition
- Set definition
- Family of sets definition
Where this leads
393 pages build on this one.
- Closure is the set of limits of sequences
- Finite intersection property
- Prior probability
- Compact set
- Second-countable space
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Chapman–Kolmogorov equation
- Totally disconnected space
- Urysohn's lemma
- Equivalence classes form a partition
- Isolated point
- Cantor set is closed
- Open set
- Open balls form a basis of the metric topology
- Closure is closed
- Arzelà–Ascoli theorem
- Law of total variance
- Equivalence of norms in finite dimension
- Torus
- Lebesgue number lemma
- Continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Möbius strip
- Law of total expectation
- Zorn's lemma
- Product topology on an arbitrary product
- Compact iff complete and totally bounded
- Union of a family of sets
- T1 space
- Homeomorphism
- Groups of prime order are cyclic
- Composition of continuous maps is continuous
- Inclusion of a subspace is continuous
- Closed subsets of compact sets are compact
- Closure in a metric space
- Heine–Cantor theorem
- Basis of a topology
- Correspondence between equivalence relations and partitions
- Continuity at a point of a topological space
- Heine–Borel theorem
- Absolutely convergent series in a Banach space converge
- Intersection of a family of sets
- Sequence spaces lp
- Topological embedding
- Order of an element divides the order of the group
- Comparison of topologies
- Riesz's lemma
- Intermediate value theorem for connected spaces
- Fixed points of a p-group action
- Partition of a set
- Open sets characterised by neighbourhoods
- Second Sylow theorem
- Closed sets contain their limits
- Completeness of the space of bounded continuous functions
- Groups of order p squared are abelian
- Metrizable space
- Finite intersection of open sets is open
- Intersection of closed sets is closed
- The number of Sylow subgroups is the index of the normaliser
- Product of compact sets is compact
- Open sets of a metric space
- Metric topology
- Baire category theorem
- Finite-dimensional normed spaces are complete
- Metrizable spaces are normal
- Normal space
- Open balls are open
- Continuity via closed sets
- Connected subsets of the real line are intervals
- Banach space
- Linear maps on finite-dimensional spaces are continuous
- Interior
- Sequential compactness
- Subbasis
- Topological space
- Box topology
- Dense set
- A set is closed iff it equals its closure
- Baire space
- Subspace topology
- Limits are unique in Hausdorff spaces
- Connected space
- Closure
- Base rate fallacy
- Law of total probability
- Epsilon-delta continuity in metric spaces
- Continuous image of a connected set is connected
- Locally compact space
- Cantor's intersection theorem
- Bounded linear maps are continuous
- Closed subsets of complete spaces are complete
- Universal property of the quotient topology
- Meagre set
- Completion of a metric space
- Closed set
- Separable space
- The alternating group A5 is simple
- Interior is the largest open subset
- Limit point
- Union of open sets is open
- First-countable space
- Axiom of choice
- Equivalent norms
- Regular space
- Boundary
- Abelian simple groups have prime order
- Cauchy's theorem
- Tychonoff's theorem
- Bayesian inference
- Path-connected space
- Topologist's sine curve
- Complement of an open set is closed
- Compactness and sequential compactness agree in metric spaces
- Continuous image of a compact set is compact
- Posterior probability
- Hausdorff maximal principle
- Nowhere dense set
- The number of Sylow subgroups divides the index
- Universal property of the product topology
- One-point compactification
- Compact Hausdorff spaces are normal
- Equicontinuity
- Hausdorff space
- Open connected subsets of R^n are path-connected
- Compact subsets of a Hausdorff space are closed
- Well-ordering theorem
- Bayes' theorem
- Connected components are closed
- Open cover
- Closed map
- Compactness via open covers
- Metric spaces are Hausdorff
- Sup norm on bounded continuous functions
- Monty Hall problem
- Converse of Lagrange's theorem fails for the alternating group A4
- Heine–Borel theorem in finite-dimensional normed spaces
- Projections are continuous
- Interior is open
- Third Sylow theorem
- Compact space
- Frattini argument
- Connected set
- T0 space
- Compact subsets of a metric space are bounded
- Cantor set is homeomorphic to the space of binary sequences
- Every infinite set has a countably infinite subset
- Size of an orbit times the size of the stabiliser
- Continuous linear map
- Path component
- Closure is the smallest closed superset
- Open map
- Continuous map of topological spaces
- Equivalent metrics
- Surjective iff it has a right inverse
- Continuous real function on a compact set attains its bounds
- Quotient topology
- First Sylow theorem
- Continuity is equivalent to sequential continuity in metric spaces
- Compact closed unit ball implies finite dimension
- Complete subspaces are closed
- Law of total probability for an event and its complement
- Gambler's ruin
- Circle as a quotient of the real line
- Countable intersection of dense open sets is dense
- Tietze extension theorem
- Absorbing Markov chain
- Comparability of cardinals
- Groups of order pq
- Continuity via open sets
- Closed balls are closed
- Choice function
- Burnside's lemma
- Proper metric space
- Product topology
- Countable union of countable sets is countable
- g to the order of the group is the identity
- Path-connected spaces are connected
- Operator norm
- Relative CW complex
- Riemann domain over C^n
- Univalent function
- Translation surface
- Bernstein–von Mises theorem
- Busemann G-space
- Stone–Weierstrass theorem
- Pettis measurability theorem
- Arakelyan's theorem
- Kirkman's schoolgirl problem
- Locally compact group
- Nonexistence of infinite-dimensional Lebesgue measure
- Köthe–Toeplitz dual
- Hypergraph
- Classification of surfaces
- Namioka--Phelps
- Lagrange's theorem
- Nilradical is the intersection of the prime ideals
- L2 convergence of Fourier series
- Ultrafilter lemma
- Brouwer's characterization of the Cantor space
- Boolean model
- Convergence of the Neumann series for Volterra kernels
- Exponential of a bounded operator
- Kolmogorov–Arnold–Moser theorem
- Schwarz–Christoffel mapping
- Series in Banach spaces
- Teichmüller space
- Galois group of a prime degree polynomial with two nonreal roots
- Minkowski question-mark function
- Céa's lemma
- Consistency of the maximum likelihood estimator
- Empirical Bayes method
- Pettis integral
- Strongly measurable function
- Brent's method
- Mergelyan's theorem
- CW complex
- High indices theorem
- E-infinity operad
- Hille–Yosida theorem
- Bayes factor
- Diaconescu's theorem
- Global attractor
- Runge's theorem
- Bump function
- Vanishing mean oscillation
- Jacobson topology
- Spectral decomposition theorem
- Banach bundle
- Absolute neighborhood retract
- Lindley's paradox
- Tychonoff space
- Kunen's inconsistency theorem
- Bayes estimator
- Bisection method
- Interior algebra
- Nichols–Zoeller theorem
- Lax–Milgram theorem
- Asymptotic normality of the maximum likelihood estimator
- Neumann series
- Standard map
- Hall's theorem on solvable groups
- Δ-complex
- Lebesgue covering dimension
- Bourbaki–Witt theorem
- Continuous functional calculus
- Hausdorff paradox
- Moore family
- Minimum total potential energy principle
- Continuous-time Markov chain
- Banach limit
- Branched covering
- Order dimension
- Carathéodory's theorem on boundary extension
- Fréchet manifold
- Parry measure
- Proof that the order of an element divides the order of the group
- Arnold's cat map
- Topological ring
- Birkhoff's subdirect representation theorem
- Suslin's problem
- Hadamard gap theorem
- Singular simplicial set
- Regular surface
- Hall subgroup
- Characterizations of semisimple modules
- Harmonic measure
- FK-space
- Anosov flow
- Bochner integral
- Midpoint convex function
- Sylow theorems
- Discontinuous additive functions via a Hamel basis
- Banach–Tarski paradox
- Approximation of a kernel by degenerate kernels
- Complex p-adic numbers
- Continuous function with divergent Fourier series
- Bayes risk
- Fredholm map
- Frattini subgroup
- Bounded sequences ℓ∞
- Sequence space
- Proof that groups of prime order are cyclic
- Krasner's lemma
- Action-angle coordinates
- Reduction of a first-kind Volterra equation to the second kind
- Space of convergent sequences
- Sinai billiard
- Liouville–Neumann series
- Topological field
- Dvoretzky–Rogers theorem
- Lusternik–Schnirelmann category
- Kronecker's approximation theorem
- Schauder fixed-point theorem for integral equations
- Milnor's isospectral flat tori
- Poincaré inequality
- Berkovich space dynamics
- Credible interval
- Conformal modulus
- Contraction mapping principle for integral equations
- Density of trigonometric polynomials in continuous functions
- Completion of a valued field
- Orbifold
- Ill-posedness of first-kind equations
- Nonexistence of a projective plane of order 10
- Spectral radius formula
- Subshift of finite type
- Fisher's inequality
- Torus knot
- Suslin line
- Nested intervals theorem
- Compactness of integral operators
- Hankel operator
- Orientation of a manifold
- Galerkin method
- Baer's criterion
- Compactness of families of harmonic functions
- Real projective plane
- Curtis–Hedlund–Lyndon theorem
- Orlicz–Pettis theorem
- Alexander subbase theorem
- Commutative Gelfand–Naimark theorem
- Jeffreys prior
- Transcendence degree
- Fourier basis of L2 of the circle
- Hyperbolic set
- Nerve of a cover
- Hilbert manifold
- Jentzsch's theorem on positive kernels
- CW approximation
- Quasisymmetric map
- Liouville–Arnold theorem
- Bayesian hierarchical model
- Banach–Stone theorem
- Gödel's consistency of CH and AC
- Banach–Mazur theorem
- Steiner system
- Normal family
- Bruck–Ryser–Chowla theorem
- Cardinal characteristics of the continuum
- Noncommutative geometry
- Schauder basis
- Axiom A
- Tsirelson space
- Conjugate prior
- Ladyzhenskaya–Babuška–Brezzi condition
- Singular homology
- Delta derivative on a time scale
- Ultrafilter
- Boolean prime ideal theorem
- Resolvent as Laplace transform of a semigroup
- Szpilrajn extension theorem
- Extremal length
- Banach algebra
- Block design
- Iterated function system
- Tame Fréchet space
- Von Neumann–Bernays–Gödel set theory
- Basel problem via Fourier series
- Banach–Lie group
- Krull topology
- Banach manifold
- BK-space
- Shrinkage estimator
- Graph embedding
- Thom–Mather stratified space
- Cantor algebra
- Convenient vector space
- Proof of Lagrange's theorem
- Existence of transcendence bases
- Particle filter
- Cantor space
- Axiom of determinacy
- Space of null sequences
- Natural boundary
- Riemann surface
- Unique solvability of Volterra equations of the second kind
- Vitali set
- Pseudo-Anosov map
- Anosov diffeomorphism
- Analytic capacity
- Mittag-Leffler's theorem
- Algorithmic randomness
- Stability conjecture
- Geometric realization
- Prime end
- Torus as a group
- Semi-continuity
- Automorphism group of a bounded domain is a Lie group
- Rellich–Kondrachov theorem
- Bunimovich stadium
- Maximal torus
- Cauchy's functional equation
- Faber polynomials
- Montel's theorem
Further reading
- Web https://en.wikipedia.org/wiki/Family_of_sets, lead section
About this page
| Tag | 05Q6G4EH2G |
|---|---|
| URI | m2://05Q6G4EH2G |
| Kind | definition |
| Writing | W1 stated |
| MSC | 03E20 |
| Defines | Family of sets |
| Uses | Set |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.