In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function...

a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is...

them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian...

topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable...

metric (FLRW), where it corresponds to an increase in the scale of the spatial part of the universe's spacetime metric tensor (which governs...

{\displaystyle X} is a discrete uniform space if it is equipped with its discrete uniformity. the discrete metric ρ {\displaystyle \rho } on X {\displaystyle...

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number...

Minkowski metric. Minkowski space is, in particular, not a metric space and not a Riemannian manifold with a Riemannian metric. However, Minkowski space contains...

Kolmogorov quotient is the one-point space. A first-countable, separable Hausdorff space (in particular, a separable metric space) has at most the continuum cardinality...

mathematics, metric may refer to one of two related, but distinct concepts: A function which measures distance between two points in a metric space A metric tensor...

pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced...

mathematics, specifically in category theory, a generalized metric space is a metric space but without the symmetry property and some other properties...

of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental,...

a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended"...

mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in...

usually agree in a metric space, but may not be equivalent in other topological spaces. One such generalization is that a topological space is sequentially...

product space Kolmogorov space Lp-space Lens space Liouville space Locally finite space Loop space Lorentz space Mapping space Measure space Metric space Minkowski...

analogue of an n-sphere (with its canonical Riemannian metric). The main application of de Sitter space is its use in general relativity, where it serves as...

mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space ( X , τ ) {\displaystyle (X...

analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation...

In mathematics, a dilation is a function f {\displaystyle f} from a metric space M {\displaystyle M} into itself that satisfies the identity d ( f ( x...

space allows defining distances and angles there. More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at...

not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called...

mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications...

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other...

These definitions coincide for subsets of a complete metric space, but not in general. A metric space ( M , d ) {\displaystyle (M,d)} is totally bounded...

or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space M {\displaystyle M} . It is...

the k closest points. Most commonly M is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle...

detailed expositions of the definitions given below. Connection Curvature Metric space Riemannian manifold See also: Glossary of general topology Glossary of...

called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian...