What (who) is complete metric space - definition
Complete metric space
METRIC SPACE IN WHICH CAUCHY SEQUENCE CONVERGES TO AN ELEMENT OF THE SPACE
Complete metric; Completeness (topology); Completion (topology); Complete (topology); Cauchy complete; Cauchy complete space; Cauchy completion; Cauchy completeness; Completion (metric space); Banach completion; Completeness (in topology); Complete space; Metric space completion; Completion of a metric space; Complete pseudometric space; Complete pseudometric
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
complete metric space
METRIC SPACE IN WHICH CAUCHY SEQUENCE CONVERGES TO AN ELEMENT OF THE SPACE
Complete metric; Completeness (topology); Completion (topology); Complete (topology); Cauchy complete; Cauchy complete space; Cauchy completion; Cauchy completeness; Completion (metric space); Banach completion; Completeness (in topology); Complete space; Metric space completion; Completion of a metric space; Complete pseudometric space; Complete pseudometric
<theory> A metric space in which every sequence that
converges in itself has a limit. For example, the space of
real numbers is complete by Dedekind's axiom, whereas the
space of rational numbers is not - e.g. the sequence a[0 ]=1;
a[n_+1 ]:=a[n ]/2+1/a[n ].
(1998-07-05)
Metric space
![[[Euler diagram]] of types of functions between metric spaces.](https://commons.wikimedia.org/wiki/Special:FilePath/Functions between metric spaces.svg?width=200 "[[Euler diagram]] of types of functions between metric spaces.")
SET EQUIPPED WITH A METRIC (DISTANCE FUNCTION)
Distance function; Metric spaces; Metric topology; Metric geometry; Bounded metric space; Bounded metric; Bounded space; Metric (mathematics); Quasimetric space; Translation-invariant metric; Translation invariant metric; Semi-metric; Semi metric space; Homogeneous metric; Quotient metric space; Semimetric; Postoffice metric; Post office metric; British rail metric; SNCF metric; Quasimetric; Quasi-metric; Premetric space; Prametric; Hemimetric space; Hemimetric; Semimetric space; Hemi-metric; Prametrizable; Distance to a set; Metric distance; General metric; Relation of norms and metrics; Inframetric; Post Office metric; Prametric space; Premetric; Finite metric space; Discrete metric space; Metric Geometry; Distance metric; Pseudoquasimetric space; Pseudo-semimetric; French railway metric; British Rail metric; Norm induced metric; Diameter of a metric space; Generalizations of metric spaces; Metric embeddings
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 called a metric or distance function.
