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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
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
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
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
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
\|\mathbf {u} -\mathbf {v} \|.} This turns the seminormed space into a pseudometric space (notice this is weaker than a metric) and allows the definition
Normed_vector_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
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
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
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
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
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
Topological space with a notion of uniform properties
single pseudometric f . {\displaystyle f.} Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces. For a
Uniform_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
Sierpiński_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
Σ-algebra
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
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
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
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
Generalization of a sequence of points
inclusion. Suppose ( M , d ) {\displaystyle (M,d)} is a metric space (or a pseudometric space) and M {\displaystyle M} is endowed with the metric topology
Net_(mathematics)
Smooth manifold with an inner product on each tangent space
g {\displaystyle d_{g}} , called the geodesic distance, is always a pseudometric (a metric that does not separate points), but it may not be a metric
Riemannian_manifold
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
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
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
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
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
Metric geometry
Ordered topological vector space Pseudometric space – Generalization of metric spaces in mathematics Uniform space – Topological space with a notion of uniform
Generalised_metric
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
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
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 ∈ a n M } . {\displaystyle d(x,y)=2^{-\sup
I-adic_topology
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
{\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
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 mathematics
generally, this remains true whenever the space is a topological group or the topology is defined by a pseudometric. Suppose u ∈ U ⊆ X {\displaystyle u\in
Neighbourhood_system
Concept in mathematical set theory
Hence, 1° ⇒ 3°. ◼ {\displaystyle \qquad \blacksquare } In a pseudometric proximal relator space X {\displaystyle X} , the neighbourhood of a point x ∈ X
Near_sets
Concepts in probability mathematics
convergence on that topology. This topology is defined by the family of pseudometrics { ρ F : F ∈ Σ , μ ( F ) < ∞ } , {\displaystyle \{\rho _{F}:F\in \Sigma
Convergence_in_measure
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
sufficient conditions for metrizability and orderability of pseudometric and ultrametric spaces. "Papić, Pavle". Croatian Encyclopedia (in Croatian). Miroslav
Pavle_Papić
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
Distance function
_{K}\lambda w\},\quad m(v/w)=\sup\{\mu :\mu w\leq _{K}v\}.} The Hilbert pseudometric on K ∖{0} is then defined by the formula d ( v , w ) = log M ( v /
Hilbert_metric
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
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)
{\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
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
Class of distance functions defined between probability distributions
D_{\mathcal {F}}(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
Concept in probability theory
{\displaystyle e=(e(x),0\leq x\leq 1)} be a Brownian excursion. Define a pseudometric d {\displaystyle d} on [ 0 , 1 ] {\displaystyle [0,1]} with d ( x , y
Brownian_tree
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
Carathéodory_metric
Sudeb (2003). "Schwarz's lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds". In Komori, Y.; Markovic, V.; Series, C
Complex_geodesic
PSEUDOMETRIC SPACE
PSEUDOMETRIC SPACE
Boy/Male
Hindu
Space
Girl/Female
Indian, Telugu
Space
Boy/Male
Hindu
Space
Girl/Female
Tamil
Antariksha | அஂதரிகà¯à®·
Space, Sky
Antariksha | அஂதரிகà¯à®·
Boy/Male
Muslim
Open space, Battle field
Boy/Male
Biblical
Breadth, space, extent.
Boy/Male
Indian
Open space, Battle field
Surname or Lastname
English or Scottish
English or Scottish : unexplained.
Girl/Female
Indian, Japanese, Tamil
Space; Star
Boy/Male
Arabic, Muslim, Pashtun
Battle Field; Open Space
Boy/Male
Hindu
Limitless space Avatar incarnation
Boy/Male
Tamil
Limitless space Avatar incarnation
Boy/Male
Hindu
Space
Girl/Female
Indian, Telugu
Goddess of Space
Surname or Lastname
English
English : occupational name for a wattler, Middle English watelere, i.e. someone who made the panels of interwoven twigs that were used to fill the spaces between the structural timbers of a timber frame building. See also Dauber.
Girl/Female
Gujarati, Hindu, Indian
Star in Space
Girl/Female
Maori
Open spaces.
Surname or Lastname
English
English : habitational name from either of two places in Cheshire. It is possible that the name originally denoted a building where village assemblies were held, named in Old English as ‘meeting-house’, from (ge)mÅt ‘meeting’ + ærn ‘house’, ‘hall’. Other possibilities are that the name derives from Old English (ge)mÅt-rÅ«m ‘meeting space’, or (ge)mÅt-treum ‘assembly trees’.
Boy/Male
Tamil
Antrix | அஂதà¯à®°à¯€à®•à¯à®·
Space
Antrix | அஂதà¯à®°à¯€à®•à¯à®·
Girl/Female
Biblical
Spaces, places.
PSEUDOMETRIC SPACE
PSEUDOMETRIC SPACE
Boy/Male
Chinese Scottish Shakespearean
Wind.
Girl/Female
Greek
Name of a woman who gave her life to save her hushand.
Boy/Male
Tamil
Lalitendu | லாலீதேஂதà¯
Beautiful Moon
Boy/Male
Muslim/Islamic
Fortunate Happy
Girl/Female
Welsh
Fair; blessed.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Traditional
Lord Shiva
Boy/Male
Tamil
Born of fire
Male
English
Anglicized form of Hebrew Shephatyah, SHEPHATIAH means "whom Jehovah defends." In the bible, this is the name of many characters, including a son of David.Â
Boy/Male
French Gaelic English
Strong.
Boy/Male
Hindu, Indian, Sanskrit
Divine; Transcendental
PSEUDOMETRIC SPACE
PSEUDOMETRIC SPACE
PSEUDOMETRIC SPACE
PSEUDOMETRIC SPACE
PSEUDOMETRIC SPACE
n.
The circular membrane that partially incloses the space beneath the umbrella of hydroid medusae.
n.
To arrange or adjust the spaces in or between; as, to space words, lines, or letters.
a.
Having the inner part cut away, or left vacant, a narrow border being left at the sides, the tincture of the field being seen in the vacant space; -- said of a charge.
a.
Alt. of Pedometrical
n.
An empty space; a vacuum.
n.
The space inclosed between ranges of hills or mountains; the strip of land at the bottom of the depressions intersecting a country, including usually the bed of a stream, with frequently broad alluvial plains on one or both sides of the stream. Also used figuratively.
n.
A quantity or portion of extension; distance from one thing to another; an interval between any two or more objects; as, the space between two stars or two hills; the sound was heard for the space of a mile.
n.
A space entirely devoid of matter (called also, by way of distinction, absolute vacuum); hence, in a more general sense, a space, as the interior of a closed vessel, which has been exhausted to a high or the highest degree by an air pump or other artificial means; as, water boils at a reduced temperature in a vacuum.
n.
Space unfilled or unoccupied, or occupied with an invisible fluid only; emptiness; void; vacuum.
n.
A small air cell, or globular space, in the interior of organic cells, either containing air, or a pellucid watery liquid, or some special chemical secretions of the cell protoplasm.
n.
One who holds the doctrine that the space between the bodies of the universe, or the molecules and atoms of matter., is a vacuum; -- opposed to plenist.
n.
Rate of motion; the relation of motion to time, measured by the number of units of space passed over by a moving body or point in a unit of time, usually the number of feet passed over in a second. See the Note under Speed.
n.
Intermission of judicial proceedings; the space of time between the end of one term and the beginning of the next; nonterm; recess.
n.
Dimensions; compass; space occupied, as measured by cubic units, that is, cubic inches, feet, yards, etc.; mass; bulk; as, the volume of an elephant's body; a volume of gas.
n.
A border, limit, or boundary of a space; an edge, margin, or brink of something definite in extent.
n.
That which is near, or not remote; that which is adjacent to anything; adjoining space or country; neighborhood.
imp. & p. p.
of Space
a.
Without space.
n.
A forest officer appointed to walk over a certain space for inspection; a forester.
n.
A waste region; boundless space; immensity.