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...

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...

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...

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...

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...

vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable...

completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate...

completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric d on X such that...

{\textstyle \bigcap _{k=0}^{\infty }C_{k}}} and the proof is complete. ∎ In a complete metric space, the following variant of Cantor's intersection theorem...

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...