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Topological space with a dense countable subset
In 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
Separable_space
Type of vector space in math
the state space is likewise assumed to be a separable complex Hilbert space. However, it is sometimes argued that non-separable Hilbert spaces are also
Hilbert_space
Topological space whose topology has a countable base
second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T
Second-countable_space
Concept in topology
mathematics, a Polish space is a separable, completely metrizable topological space; i.e., a space homeomorphic to a complete metric space that has a countable
Polish_space
Algebraic structure of set algebra
higher than continuum). A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two
Σ-algebra
Normed vector space that is complete
a separable Banach space need not be separable, but: Theorem—Let X be a normed space. If X′ is separable, then X is separable. When X′ is separable, the
Banach_space
Topics referred to by the same term
roots is equal to its degree Separable sigma algebra, a separable space in measure theory Separable space, a topological space that contains a countable
Separability
is a zero-dimensional compact Hausdorff space. It is a linearly ordered topological space that is separable but not second countable, hence not metrizable;
Split_interval
Vector space of functions in mathematics
p}(\Omega )} is a Banach space. For p < ∞ , W k , p ( Ω ) {\displaystyle p<\infty ,W^{k,p}(\Omega )} is also a separable space. It is conventional to denote
Sobolev_space
Topological space where each point has a countable neighbourhood basis
space – Type of topological space Second-countable space – Topological space whose topology has a countable base Separable space – Topological space with
First-countable_space
Geometric property of a pair of sets of points in Euclidean geometry
higher-dimensional Euclidean spaces if the line is replaced by a hyperplane. The problem of determining if a pair of sets is linearly separable and finding a separating
Linear_separability
Right continuous function with left limits
\sigma _{0}} , D {\displaystyle \mathbb {D} } is a separable space. Thus, Skorokhod space is a Polish space. By an application of the Arzelà–Ascoli theorem
Càdlàg
Frequently-cited counterexample in topology
subset of this space, and this is a non-separable subset of the separable space S {\displaystyle \mathbb {S} } . It shows that separability does not inherit
Sorgenfrey_plane
space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces,
Weakly_measurable_function
topology, a cosmic space is any topological space that is a continuous image of some separable metric space. Equivalently (for regular T1 spaces but not in general)
Cosmic_space
Quantum states that are not entangled
In quantum mechanics, separable states are multipartite quantum states that can be written as a convex combination of product states. Product states are
Separable_state
Measure of the shape of a function
that M is a separable space with respect to the metric d.) Let 1 ≤ p ≤ ∞. The pth central moment of a measure μ on the measurable space (M, B(M)) about
Moment_(mathematics)
About polynomials in several variables
2026), On endomorphisms of affine spaces and the Jacobian problem, arXiv:2609.05746 Adjamagbo, Kossivi (1995), "On separable algebras over a U.F.D. and the
Jacobian_conjecture
Topological space that is homeomorphic to a metric space
Hausdorff space is metrizable if and only if it is second-countable. Urysohn's Theorem can be restated as: A topological space is separable and metrizable
Metrizable_space
Type of algebraic field extension
a separable extension if for every α ∈ E {\displaystyle \alpha \in E} , the minimal polynomial of α {\displaystyle \alpha } over F is a separable polynomial
Separable_extension
Collection of random variables
For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space, which means that the index set has
Stochastic_process
Vector space with generalized dot product
Any separable inner product space has an orthonormal basis. Using the Hausdorff maximal principle and the fact that in a complete inner product space orthogonal
Inner_product_space
Type of relation for subsets of a topological space
should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological
Separated_sets
Mathematical theorem related to real and functional analysis
be found in (Schwartz 1969, p. 21). If H1 is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory;
Kirszbraun_theorem
Vector space with a notion of nearness
separated if it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together
Topological_vector_space
Topological space of dimension zero
above agree for separable, metrisable spaces (see Inductive dimension § Relationships between dimensions). A zero-dimensional Hausdorff space is necessarily
Zero-dimensional_space
Branch of topology
separable, and Lindelöf. Every σ-compact space is Lindelöf. A metric space is first-countable. For metric spaces second-countability, separability, and
General_topology
Vector space of infinite sequences
construction of Tsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to a quotient space of ℓ 1 {\displaystyle
Sequence_space
Mathematical construction in topology
makes it a complete separable metric space in such a way that Σ {\displaystyle \Sigma } is then the Borel σ-algebra. Standard Borel spaces have several useful
Standard_Borel_space
In mathematics, the Urysohn universal space is a certain metric space that contains all separable metric spaces in a particularly nice manner. This concept
Urysohn_universal_space
Index of articles associated with the same name
space is sequential. Every second-countable space is first countable, separable, and Lindelöf. Every σ-compact space is Lindelöf. Every metric space is
Axiom_of_countability
Algorithm for supervised learning of binary classifiers
them into a binary space. In fact, for a projection space of sufficiently high dimension, patterns can become linearly separable. Another way to solve
Perceptron
Polish mathematician (1907–1976)
Marczewski proved that the topological dimension, for arbitrary metrisable separable space X, coincides with the Hausdorff dimension under one of the metrics
Edward_Marczewski
space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable and complete metric space. Polyadic A space is
Glossary_of_general_topology
Mathematical set with some added structure
is closed in the product space. Every Borel set in a Euclidean space (and more generally, in a complete separable metric space), endowed with the Borel
Space_(mathematics)
Type of mathematical space
second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies
Compact_space
on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure space (X
Abelian_von_Neumann_algebra
Generalization of compactness
Arzelà–Ascoli theorem. A metric space is separable if and only if it is homeomorphic to a totally bounded metric space. The closure of a totally bounded
Totally_bounded_space
Technique for solving differential equations
differential equation for the unknown f ( x ) {\displaystyle f(x)} is separable if it can be written in the form d d x f ( x ) = g ( x ) h ( f ( x ) )
Separation_of_variables
Type of topological space
countable, dense subset is called a separable space. For a non-compact, locally compact Hausdorff topological space ( X , τ X ) {\displaystyle (X,\tau
Polyadic_space
Function spaces generalizing finite-dimensional p norm spaces
infinite, this is exactly the sequence space ℓp defined above. For uncountable sets I this is a non-separable Banach space which can be seen as the locally
Lp_space
certain well-behaved normed spaces (separable). It states that every such normed space can be embedded into the normed space C ( [ 0 , 1 ] , R ) {\displaystyle
Banach–Mazur_theorem
Metric geometry
topological spaces, the completely uniformizable spaces. A topological space homeomorphic to a separable complete metric space is called a Polish space. Since
Complete_metric_space
Topological vector spaces
space. The space D ′ ( U ) {\displaystyle {\mathcal {D}}^{\prime }(U)} is separable and has the strong Pytkeev property but it is neither a k-space nor
Spaces of test functions and distributions
Spaces_of_test_functions_and_distributions
Generalized notion of measure in mathematics
theorem. If X is a compact separable space, then the space of finite signed Baire measures is the dual of the real Banach space of all continuous real-valued
Signed_measure
In mathematics, vector space of linear forms
normed space V {\displaystyle V} is separable, then so is the space V {\displaystyle V} itself. The converse is not true: for example, the space l 1 {\displaystyle
Dual_space
Statement in computational learning theory
high-dimensional space nonlinearly, is more likely to be linearly separable than in a low-dimensional space, provided that the space is not densely populated
Cover's_theorem
Locally convex topological vector space that is also a complete metric space
Brauner spaces. All metrizable Montel spaces are separable. A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in
Fréchet_space
Condition in order theory and topology
Every separable topological space has the ccc. Furthermore, a product space of an arbitrary number of separable spaces has the ccc. A metric space has the
Countable_chain_condition
Problem in set theory
requirement that R contains a countable dense subset (i.e., R is a separable space), then the answer is indeed yes: any such set R is necessarily order-isomorphic
Suslin's_problem
Space with topology generated by convex sets
argument for the case of separable normed spaces (in which case the unit ball of the dual is metrizable). Suppose X is a vector space over 𝕂, a subfield of
Locally convex topological vector space
Locally_convex_topological_vector_space
Type of topological space
a subspace of the real line is the discrete topology. A discrete space is separable if and only if it is countable. Any topological subspace of R {\displaystyle
Discrete_space
Concept in mathematics
second-countable space (it has a countable base of open sets) is a separable space (it has a countable dense subset). A metric space is separable if and only
Axiom_of_countable_choice
{\displaystyle X} be an infinite-dimensional closed subspace of a separable Orlicz sequence space ℓ M {\displaystyle \ell _{M}} . Then X {\displaystyle X} has
Orlicz_sequence_space
Combinatorial and geometric result used in measure theory of Euclidean spaces
{F} } be an arbitrary collection of non-degenerating balls in a separable metric space such that R := sup { r a d ( B ) : B ∈ F } < ∞ {\displaystyle R:=\sup
Vitali_covering_lemma
Theorem in functional analysis
space of C([a,b]) is countably weak-* compact. In 1932, Stefan Banach proved that the closed unit ball in the continuous dual space of any separable normed
Banach–Alaoglu_theorem
Space of stochastic processes
to the uniform metric, C {\displaystyle C} is both a separable and a complete space: Separability is a consequence of the Stone–Weierstrass theorem; Completeness
Classical_Wiener_space
Topological space which is a generalization of certain compact spaces
Rudin. Existing proofs of this require the axiom of choice for the non-separable case. It has been shown that ZF theory is not sufficient to prove it,
Paracompact_space
Topology where a set is open if it contains a particular point
separated sets. Separability {p} is dense and hence X is a separable space. However if X is uncountable then X \ {p} is not separable. This is an example
Particular_point_topology
Two-stage small launch vehicle
Spaceport in Norway and the Guiana Space Centre in French Guiana. Early customers for the launcher include Airbus Defence and Space, the German Aerospace Center
Spectrum_(rocket)
Notion in measure theory
gives the general flavor. Theorem. Suppose X is a Polish space and Y a separable Hausdorff space, both equipped with their Borel σ-algebras. Let μ be a
Lifting_theory
Type of topological space
example, there are many compact spaces that are not second-countable. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable
Lindelöf_space
Topological space characterized by sequences
Shou, Lin; Chuan, Liu; Mumin, Dai (1997). "Images on locally separable metric spaces". Acta Mathematica Sinica. 13 (1): 1–8. doi:10.1007/BF02560519
Sequential_space
Mathematical folklore
infinite-dimensional spaces due to a key limitation: any translation-invariant Borel measure on an infinite-dimensional separable Banach space must be either
Infinite-dimensional Lebesgue measure
Infinite-dimensional_Lebesgue_measure
Topological vector space
every LF-space is ultrabornological. An LF-space that is the inductive limit of a countable sequence of separable spaces is separable. LF spaces are distinguished
LF-space
Locally convex topological vector space
reflexive Banach space is separable if and only if its continuous dual is separable. This follows from the fact that for every normed space Y , {\displaystyle
Reflexive_space
Partially unsolved problem in mathematics
subspaces is an operator that acts on a Banach space that is not isomorphic to a separable Hilbert space). The problem seems to have been stated in the
Invariant_subspace_problem
In functional analysis, a Hilbert space
Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H {\displaystyle
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
space Locally compact space Compactly generated space Axiom of countability Sequential space First-countable space Second-countable space Separable space
List of general topology topics
List_of_general_topology_topics
Non-separable wavelets are multi-dimensional wavelets that are not directly implemented as tensor products of wavelets on some lower-dimensional space. They
Non-separable_wavelet
Used to compare mixed characteristic situations with purely finite characteristic ones
étale over K♭. Since finite étale maps into a field are exactly finite separable field extensions, the almost purity theorem implies that for any perfectoid
Perfectoid_space
Topological space
topology. Thus, the Moore plane shows that a subspace of a separable space need not be separable. The Moore plane is first countable, but not second countable
Moore_plane
All infinite-dimensional, separable Banach spaces are homeomorphic
two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are homeomorphic as topological spaces. The theorem was proved by
Anderson–Kadec_theorem
Mathematical space representing physical quantum systems
phase space of classical mechanics. In quantum mechanics a state space is a separable complex Hilbert space. The dimension of this Hilbert space depends
Quantum_state_space
Mathematical concept
{\hat {A}}\cong \operatorname {Prim} (A).} Let H be a separable infinite-dimensional Hilbert space. L(H) has two norm-closed *-ideals: I0 = {0} and the
Spectrum_of_a_C*-algebra
Construct in quantum information theory
range of possible expectation values of any separable state. Let a composite quantum system have state space H A ⊗ H B {\displaystyle H_{A}\otimes H_{B}}
Entanglement_witness
plane) is an example of a non-metrizable Moore space. Every metacompact, separable, normal Moore space is metrizable. This theorem is known as Traylor's
Moore_space_(topology)
Topological space in mathematics
has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as an important counterexample in topology. Intuitively
Long_line_(topology)
Mathematical expression for linear operators
a product of separable polynomials. Let x : V → V {\displaystyle x:V\to V} be any linear operator on the finite-dimensional vector space V {\displaystyle
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Branch of mathematical logic
are restricted to separable spaces. Many principles that imply the axiom of choice in their general form (such as "Every vector space has a basis") become
Reverse_mathematics
Real function with finite total variation
Fomin (1969, example 7, pp. 48–49), in order to prove the non separability of the space of bounded sequences, and also Kannan & Krueger (1996, example
Bounded_variation
Theorem in measure theory
Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations
Prokhorov's_theorem
FK-AK spaces are separable spaces. BK-space – Sequence space that is Banach FK-space – Sequence space that is Fréchet Normed space – Vector space on which
FK-AK_space
holomorphic separability is a measure of the richness of the set of holomorphic functions on a complex manifold or complex-analytic space. A complex manifold
Holomorphic_separability
Matrix decomposition
𝐌 on a separable Hilbert space H. Namely, for any bounded operator 𝐌, there exist a partial isometry 𝐔, a unitary 𝐕, a measure space (X,μ)
Singular_value_decomposition
Functional analysis concept
corresponding eigenvalues λn∈ℝ, such that λn→0. When the Hilbert space is in addition separable, one can mix the basis (en) with a countable orthonormal basis
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
mathematical logic, an effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in effective descriptive
Effective_Polish_space
Mathematical theorem
abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space. It was proved
Structure theorem for Gaussian measures
Structure_theorem_for_Gaussian_measures
Topological vector space whose topology can be defined by a metric
infinite-dimensional separable complete metrizable TVS are homeomorphic. Every metrizable locally convex TVS is a quasibarrelled space, bornological space, and a Mackey
Metrizable topological vector space
Metrizable_topological_vector_space
Mathematical concept
normed space X has a dual space that is separable (with respect to the dual-norm topology) then X is necessarily separable. If X is a Banach space, the
Weak_topology
Barrelled space where closed and bounded subsets are compact
bounded. A Fréchet–Montel space is a Fréchet space that is also a Montel space. A separable Fréchet space is a Montel space if and only if each weak-*
Montel_space
Certain topology in mathematics
base is ω1. Some further properties include neither [0,ω1) or [0,ω1] is separable or second-countable [0,ω1] is compact, while [0,ω1) is sequentially compact
Order_topology
Subject in mathematics
Let ( X , T ) {\displaystyle (X,{\mathcal {T}})} be a separable, metrizable locally convex space and T s := T s ( X , X ′ ) {\displaystyle T_{s}:=T_{s}(X
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
Upper bound in coding theory
achieve equality in the Singleton bound are called MDS (maximum distance separable) codes. Examples of such codes include codes that have only q {\displaystyle
Singleton_bound
Type of continuity of a complex-valued function
{\displaystyle U} . The space C 0 , α ( Ω ) , 0 < α ≤ 1 {\displaystyle C^{0,\alpha }(\Omega ),0<\alpha \leq 1} is not separable. The embedding C 0 , β
Hölder_condition
Generalization of the concept of a direct sum in mathematics
classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous
Direct_integral
Tensor product space endowed with a special inner product
subspaces. This definition is almost never separable, in part because, in physical applications, "most" of the space describes impossible states. Modern authors
Tensor product of Hilbert spaces
Tensor_product_of_Hilbert_spaces
Mathematical construction relating to infinite-dimensional spaces
abstract Wiener space construction. Let H {\displaystyle H} be a real Hilbert space, assumed to be infinite dimensional and separable. In the physics
Abstract_Wiener_space
Mathematical concept
are only finitely many open sets) and separable (since the space itself is countable). If a finite topological space is T1 (in particular, if it is Hausdorff)
Finite_topological_space
*-algebra of bounded operators on a Hilbert space
Neumann algebra that acts on a separable Hilbert space is called separable. Note that such algebras are rarely separable in the norm topology. The von
Von_Neumann_algebra
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