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Generalization of metric spaces in mathematics
mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were
Pseudometric_space
Topological vector space whose topology can be defined by a metric
by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS. A pseudometric on a set X is a map
Metrizable topological vector space
Metrizable_topological_vector_space
Pseudometric of complex manifolds
mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by
Kobayashi_metric
Metric geometry
limit exists because the real numbers are complete.) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the
Complete_metric_space
Topological space with a notion of uniform properties
individual pseudometrics fi. If the family of pseudometrics is finite, it can be seen that the same uniform structure is defined by a single pseudometric, namely
Uniform_space
Topics referred to by the same term
Pseudometric may refer to: The metric of a pseudo-Riemannian manifold, a non-degenerate, smooth, symmetric tensor field of arbitrary signature Pseudometric
Pseudometric
Mathematical space with a notion of distance
words, the axioms for a pseudometric are: d(x,y)≥0 d(x,x)=0 d(x,y)=d(y,x) d(x,z)≤d(x,y)+d(y,z). In some contexts, pseudometrics are referred to as semimetrics
Metric_space
Structure in functional analysis
the general theory of complete pseudometric spaces. Recall that every metric is a pseudometric and that a pseudometric p {\displaystyle p} is a metric
Complete topological vector space
Complete_topological_vector_space
Quantitative way to compare statistical models
Blackwell–Sherman–Stein theorem. Closely related is the Le Cam distance, a pseudometric for the maximum deficiency between two statistical models. If the deficiency
Deficiency_(statistics)
Algebraic structure of set algebra
continuum). A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is defined
Σ-algebra
On topological spaces where the intersection of countably many dense open sets is dense
metrizable topological space is a Baire space. More generally, every complete pseudometric space is a Baire space. (BCT2) Every locally compact Hausdorff space
Baire_category_theorem
Mathematical function
the seminorm-induced topology, via the canonical translation-invariant pseudometric d p : X × X → R {\displaystyle d_{p}:X\times X\to \mathbb {R} } ; d p
Seminorm
Topology where the only open sets are the empty set and the entire space
distinguished by topological means. Every indiscrete space can be viewed as a pseudometric space in which the distance between any two points is zero. The trivial
Trivial_topology
Statistical machine learning algorithm for metric learning
statistical machine learning algorithm for metric learning. It learns a pseudometric designed for k-nearest neighbor classification. The algorithm is based
Large_margin_nearest_neighbor
Type of topological space
infinite set, as is the cocountable topology defined on an uncountable set. Pseudometric spaces typically are not Hausdorff, but they are preregular, and their
Hausdorff_space
Concept in topology
is a sensible structure on X; it is a pseudometric. (Again, there is a more direct definition of pseudometric.) In this way, there is a natural way to
Kolmogorov_space
Distance between two statistical objects
because they lack one or more properties of proper metrics. For example, pseudometrics violate property (2), identity of indiscernibles; quasimetrics violate
Statistical_distance
Space with topology generated by convex sets
that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is
Locally convex topological vector space
Locally_convex_topological_vector_space
Type of topological space
(and hence all metrizable spaces) are perfectly normal Hausdorff; All pseudometric spaces (and hence all pseudometrizable spaces) are perfectly normal regular
Normal_space
generated by a partition P {\displaystyle P} can be viewed as a pseudometric space with a pseudometric given by: d ( x , y ) = { 0 if x and y are in the same
Partition_topology
Elements in exactly one of two sets
= μ ( X Δ Y ) {\displaystyle d_{\mu }(X,Y)=\mu (X\,\Delta \,Y)} is a pseudometric on Σ. dμ becomes a metric if Σ is considered modulo the equivalence relation
Symmetric_difference
Vector space on which a distance is defined
vectors 𝐮 and 𝐯 as ‖𝐮−𝐯‖. This turns the seminormed space into a pseudometric space (notice this is weaker than a metric) and allows the definition
Normed_vector_space
Distance between two metric-space subsets
intersects Y. On the set of all subsets of M, dH yields an extended pseudometric. On the set F(M) of all non-empty compact subsets of M, dH is a metric
Hausdorff_distance
Concept in topology
conditions for a topological space to be a Baire space. (BCT1) Every complete pseudometric space is a Baire space. In particular, every completely metrizable topological
Baire_space
Type of regular Hausdorff space
topology. Other examples include: Every metric space is Tychonoff; every pseudometric space is completely regular. Every locally compact regular space is completely
Tychonoff_space
continuous function on the space is bounded. Pseudometric See Pseudometric space. Pseudometric space A pseudometric space (M, d) is a set M equipped with a
Glossary_of_general_topology
Topological space whose topology is generated by a uniform structure
induced by a family of pseudometrics; indeed, this is because any uniformity on a set X can be defined by a family of pseudometrics. Showing that a space
Uniformizable_space
Finite topological space with two points, only one of which is closed
Sierpiński space S is not metrizable or even pseudometrizable since every pseudometric space is completely regular but the Sierpiński space is not even regular
Sierpiński_space
Smooth manifold with an inner product on each tangent space
The distance function dg, called the geodesic distance, is always a pseudometric (a metric that does not separate points), but it may not be a metric
Riemannian_manifold
Concept in mathematics
whenever the space is a topological group or the topology is defined by a pseudometric. Suppose u∈U⊆X and let 𝒩 be a neighbourhood basis for u in X. Make 𝒩
Neighbourhood_system
Generalization of a sequence of points
partially ordered by inclusion. Suppose (M,d) is a metric space (or a pseudometric space) and M is endowed with the metric topology. If m∈M is a point and
Net_(mathematics)
Mathematical concept
if it is R0. The uniform structure will be the pseudometric uniformity induced by the above pseudometric. Perhaps surprisingly, there are finite topological
Finite_topological_space
Type of smooth complex surface of kodaira dimension 0
Kamenova, Ljudmila; Lu, Steven; Verbitsky, Misha (2014), "Kobayashi pseudometric on hyperkähler manifolds", Journal of the London Mathematical Society
K3_surface
Concept in commutative algebra
determines a topology on M called the 𝔞-adic topology, characterized by the pseudometric d(x,y)=2−sup{n∣x−y∈𝔞nM}. The family {x+𝔞nM:x∈M,n∈ℤ+} is a basis for
I-adic_topology
Topological space that is homeomorphic to a metric space
uniform space, or equivalently the topology being defined by a family of pseudometrics Simon, Jonathan. "Metrization Theorems" (PDF). Retrieved 16 June 2016
Metrizable_space
Concept in probability theory
(Xt)t∈T be a Gaussian process centered (with mean zero) and let dX be the pseudometric on T defined by d X ( s , t ) = E [ | X s − X t | 2 ] . {\displaystyle
Dudley's_theorem
Topological relational characteristic
indiscrete space, any two points are topologically indistinguishable. In a pseudometric space, two points are topologically indistinguishable if and only if
Topological indistinguishability
Topological_indistinguishability
Measure of distance between persistence modules
distance. These two properties make the interleaving distance an extended pseudometric, which means non-identical objects are allowed to have distance zero
Interleaving_distance
Analysis of datasets using techniques from topology
have been made on persistence homology with torsion. Frosini defined a pseudometric on this specific module and proved its stability. One of its novelty
Topological_data_analysis
Concepts in probability mathematics
convergence on that topology. This topology is defined by the family of pseudometrics {ρF:F∈Σ, μ(F)<∞}, where ρF(f,g)=∫Fmin{|f−g|,1}dμ. In general, one may
Convergence_in_measure
Concept in mathematical set theory
also). Let ρ X , ρ Y {\displaystyle \rho _{X},\rho _{Y}} be extended pseudometrics on nonempty sets X , Y {\displaystyle X,Y} , respectively. The map f
Near_sets
Generalization of a positive-definite matrix
{\displaystyle K=(K_{n})^{n}} . Another link is that a p.d. kernel induces a pseudometric, where the first constraint on the distance function is loosened to allow
Positive-definite_kernel
Concept in probability theory
transformation. Let e=(e(x),0≤x≤1)be a Brownian excursion. Define a pseudometric d on [0,1] with d(x,y):=e(x)+e(y)−2min{e(z);z∈[x,y]}, for any x,y∈[0
Brownian_tree
Class of distance functions defined between probability distributions
that we may have Dℱ(P,Q)=0 for some P ≠ Q; this is variously termed a "pseudometric" or a "semimetric" depending on the community. For instance, using the
Integral_probability_metric
Sudeb (2003). "Schwarz's lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds". In Komori, Y.; Markovic, V.; Series, C
Carathéodory_metric
Metric geometry
H)\subseteq H.} This completes the proof. Ordered topological vector space Pseudometric space – Generalization of metric spaces in mathematics Uniform space –
Generalised_metric
Distance function
first defines M(v/w)=inf{λ:v≤Kλw},m(v/w)=sup{μ:μw≤Kv}. The Hilbert pseudometric on K ∖{0} is then defined by the formula d(v,w)=logM(v/w)m(v/w). It
Hilbert_metric
definite. This covariance defines a semi-inner product as well as a pseudometric on L P 2 ( S ) {\displaystyle L_{P}^{2}(S)} given by ϱ P ( f , g ) =
Pregaussian_class
Concept in mathematical topology
sequence follows from the hemicompactness of X {\displaystyle X} ). Define pseudometrics d n ( f , g ) = sup x ∈ K n δ ( f ( x ) , g ( x ) ) , f , g ∈ C ( X
Hemicompact_space
necessary and sufficient conditions for metrizability and orderability of pseudometric and ultrametric spaces. "Papić, Pavle". Croatian Encyclopedia (in Croatian)
Pavle_Papić
{\displaystyle {\mathcal {G}}=({\mathcal {V}},{\mathcal {E}})} with pseudometric on the node set V {\displaystyle {\mathcal {V}}} written a i j {\displaystyle
Algebraic_signal_processing
{\displaystyle x,y\in [a,b]} , x ≤ y {\displaystyle x\leq y} , define a pseudometric and an equivalence relation with: d e ( x , y ) := e ( x ) + e ( y )
Real_tree
Sudeb (2003). "Schwarz's lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds". In Komori, Y.; Markovic, V.; Series, C
Complex_geodesic
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