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...

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...

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...

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...

From a categorical point of view, the extended pseudometric spaces and the extended pseudoquasimetric spaces, along with their corresponding nonexpansive...

non-degenerate, smooth, symmetric tensor field of arbitrary signature Pseudometric space, a generalization of a metric that does not necessarily distinguish...

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...

A space is pseudocompact if every real-valued continuous function on the space is bounded. Pseudometric See Pseudometric space. Pseudometric space A pseudometric...

(and hence all metrizable spaces) are perfectly normal Hausdorff; All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular...

\|\mathbf {u} -\mathbf {v} \|.} This turns the seminormed space into a pseudometric space (notice this is weaker than a metric) and allows the definition...

set, as is the cocountable topology defined on an uncountable set. Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use...

(BCT1) Every complete pseudometric space is a Baire space. In particular, every completely metrizable topological space is a Baire space. (BCT2) Every locally...

topological space to be a Baire space. (BCT1) Every complete pseudometric space is a Baire space. In particular, every completely metrizable topological space is...

of being homeomorphic to a uniform space, or equivalently the topology being defined by a family of pseudometrics Simon, Jonathan. "Metrization Theorems"...

include: Every metric space is Tychonoff; every pseudometric space is completely regular. Every locally compact regular space is completely regular,...

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...

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 is uniformizable...

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...

Sierpiński space S is not metrizable or even pseudometrizable since every pseudometric space is completely regular but the Sierpiński space is not even...

single pseudometric f . {\displaystyle f.} Certain authors call spaces the topology of which is defined in terms of pseudometrics gauge spaces. For a...

than continuum). A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is...

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...

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...

structure will be the pseudometric uniformity induced by the above pseudometric. Perhaps surprisingly, there are finite topological spaces with nontrivial fundamental...

sequences because unlike Fréchet spaces which are metrizable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable...

Ordered topological vector space Pseudometric space – Generalization of metric spaces in mathematics Uniform space – Topological space with a notion of uniform...

mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by...

statistical machine learning algorithm for metric learning. It learns a pseudometric designed for k-nearest neighbor classification. The algorithm is based...

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...

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...