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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
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 vector space whose topology can be defined by a metric
pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence
Metrizable topological vector space
Metrizable_topological_vector_space
Topics referred to by the same term
non-degenerate, smooth, symmetric tensor field of arbitrary signature Pseudometric space, a generalization of a metric that does not necessarily distinguish
Pseudometric
Vector space on which a distance is defined
seminormed vector space we can define the distance between two vectors đź and đŻ as âđźâđŻâ. This turns the seminormed space into a pseudometric space (notice this
Normed_vector_space
Structure in functional analysis
x=y.} Thus every metric space is a pseudometric space and a pseudometric space ( X , p ) {\displaystyle (X,p)} is a metric space if and only if p {\displaystyle
Complete topological vector space
Complete_topological_vector_space
Type of topological space
set, as is the cocountable topology defined on an uncountable set. Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use
Hausdorff_space
Mathematical space with a notion of distance
From a categorical point of view, the extended pseudometric spaces and the extended pseudoquasimetric spaces, along with their corresponding nonexpansive
Metric_space
Topology where the only open sets are the empty set and the entire space
of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space can be viewed as a pseudometric space in which
Trivial_topology
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
Topological space with a notion of uniform properties
defined by the single pseudometric f. Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces. For a family (fi)
Uniform_space
On topological spaces where the intersection of countably many dense open sets is dense
topological space is a Baire space. More generally, every complete pseudometric space is a Baire space. (BCT2) Every locally compact Hausdorff space is a Baire
Baire_category_theorem
Concept in topology
topological space to be a Baire space. (BCT1) Every complete pseudometric space is a Baire space. In particular, every completely metrizable topological space is
Baire_space
Type of regular Hausdorff space
include: Every metric space is Tychonoff; every pseudometric space is completely regular. Every locally compact regular space is completely regular,
Tychonoff_space
Algebraic structure of set algebra
than continuum). A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is
ÎŁ-algebra
Finite topological space with two points, only one of which is closed
SierpiĆski space S is not metrizable or even pseudometrizable since every pseudometric space is completely regular but the SierpiĆski space is not even
SierpiĆski_space
Topological space that is homeomorphic to a metric space
a topological space of being homeomorphic to a uniform space, or equivalently the topology being defined by a family of pseudometrics Simon, Jonathan
Metrizable_space
Concept in topology
vector space that we started with has a lot of extra structure; for example, it is a vector space, and it has a seminorm, and these define a pseudometric and
Kolmogorov_space
A space is pseudocompact if every real-valued continuous function on the space is bounded. Pseudometric See Pseudometric space. Pseudometric space A pseudometric
Glossary_of_general_topology
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 mâM is a point
Net_(mathematics)
Mathematical concept
structure will be the pseudometric uniformity induced by the above pseudometric. Perhaps surprisingly, there are finite topological spaces with nontrivial fundamental
Finite_topological_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
Space with topology generated by convex sets
sequences because unlike Fréchet spaces which are metrizable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable
Locally convex topological vector space
Locally_convex_topological_vector_space
Topological relational characteristic
always a normal subgroup). Uniform spaces generalize both pseudometric spaces and topological groups. In a uniform space, x ⥠y if and only if the pair (x
Topological indistinguishability
Topological_indistinguishability
Mathematical function
the seminorm-induced topology, via the canonical translation-invariant pseudometric d p : X Ă X â R {\displaystyle d_{p}:X\times X\to \mathbb {R} } ; d p
Seminorm
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
Ordered topological vector space Pseudometric space â Generalization of metric spaces in mathematics Uniform space â Topological space with a notion of uniform
Generalised_metric
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
Topological space whose topology is generated by a uniform structure
uniformizable spaces that are not (pseudo)metrizable. However, the topology of a uniformizable space can always be induced by a family of pseudometrics; indeed
Uniformizable_space
Quantitative way to compare statistical models
BlackwellâShermanâStein theorem. Closely related is the Le Cam distance, a pseudometric for the maximum deficiency between two statistical models. If the deficiency
Deficiency_(statistics)
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
Elements in exactly one of two sets
= ÎŒ ( X Î Y ) {\displaystyle d_{\mu }(X,Y)=\mu (X\,\Delta \,Y)} is a pseudometric on ÎŁ. dÎŒ becomes a metric if ÎŁ is considered modulo the equivalence relation
Symmetric_difference
Concept in commutative algebra
determines a topology on M called the đ-adic topology, characterized by the pseudometric d(x,y)=2âsupâĄ{nâŁxâyâđnM}. The family {x+đnM:xâM,nââ€+} is a basis for
I-adic_topology
Concept in mathematical topology
sequence follows from the hemicompactness of X {\displaystyle X} ). Define pseudometrics d n ( f , g ) = sup x â K n ÎŽ ( f ( x ) , g ( x ) ) , f , g â C ( X
Hemicompact_space
Concept in probability theory
(Xt)tâT 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 â X t | 2 ] . {\displaystyle
Dudley's_theorem
Concept in mathematics
generally, this remains true whenever the space is a topological group or the topology is defined by a pseudometric. Suppose uâUâX and let đ© be a neighbourhood
Neighbourhood_system
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
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
Concepts in probability mathematics
convergence on that topology. This topology is defined by the family of pseudometrics {ÏF:FâÎŁ, Ό(F)<â}, where ÏF(f,g)=â«FminâĄ{|fâg|,1}dÎŒ. In general, one may
Convergence_in_measure
Type of smooth complex surface of kodaira dimension 0
Kamenova, Ljudmila; Lu, Steven; Verbitsky, Misha (2014), "Kobayashi pseudometric on hyperkÀhler manifolds", Journal of the London Mathematical Society
K3_surface
Concept in mathematical set theory
Hence, 1° â 3°. ⌠{\displaystyle \qquad \blacksquare } In a pseudometric proximal relator space X {\displaystyle X} , the neighbourhood of a point x â X
Near_sets
{\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
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),0â€xâ€1)be a Brownian excursion. Define a pseudometric d on [0,1] with d(x,y):=e(x)+e(y)â2minâĄ{e(z);zâ[x,y]}, for any x,yâ[0
Brownian_tree
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
Distance function
first defines M(v/w)=infâĄ{λ:vâ€Kλw},m(v/w)=supâĄ{ÎŒ:ÎŒwâ€Kv}. The Hilbert pseudometric on K â{0} is then defined by the formula d(v,w)=logâĄM(v/w)m(v/w). It
Hilbert_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
{\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
Class of distance functions defined between probability distributions
that we may have Dâ±(P,Q)=0 for some P â Q; this is variously termed a "pseudometric" or a "semimetric" depending on the community. For instance, using the
Integral_probability_metric
sufficient conditions for metrizability and orderability of pseudometric and ultrametric spaces. "PapiÄ, Pavle". Croatian Encyclopedia (in Croatian). Miroslav
Pavle_PapiÄ
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
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
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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