Difference between revisions of "Cauchy sequence"
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''Cauchy sequence, of points in a metric space $(X,d)$'' | ''Cauchy sequence, of points in a metric space $(X,d)$'' | ||
| − | A sequence $\{ | + | A sequence $\{x_i\}$ of elements in a [[Metric space|metric space]] $(X,d)$ such that for any $\varepsilon > 0$ there is a number $N$ such that |
\[ | \[ | ||
| − | d ( | + | d (x_n, x_m) < \varepsilon \qquad \forall m,n\geq N\, . |
\] | \] | ||
| − | The latter is also called Cauchy condition. A convergent sequence is always necessarily a Cauchy sequence. However the converse is not necessarily true. A metric space with the property that any Cauchy sequence has a limit is called [[Complete metric space|complete]], see also [[Cauchy criteria]] | + | The latter is also called Cauchy condition. A convergent sequence is always necessarily a Cauchy sequence. However the converse is not necessarily true. A metric space with the property that any Cauchy sequence has a limit is called [[Complete metric space|complete]], see also [[Cauchy criteria]]. |
| − | The concept of Cauchy sequence can be generalized to Cauchy nets (see also [[Net]]; [[Net (of sets in a topological space)]], [[Generalized sequence]] and [[Cauchy filter]]) in a [[Uniform space|uniform space]]. Let $X$ be a uniform space with uniformity $\mathcal{U}$. A net $\{x_\alpha, \alpha \in A\}$ (where $A$ is a [[Directed set|directed set]] | + | The concept of Cauchy sequence can be generalized to Cauchy nets (see also [[Net]]; [[Net (of sets in a topological space)]], [[Generalized sequence]] and [[Cauchy filter]]) in a [[Uniform space|uniform space]]. Let $X$ be a uniform space with uniformity $\mathcal{U}$. A net $\{x_\alpha, \alpha \in A\}$ of elements $x_\alpha \in X$ (where $A$ is a [[Directed set|directed set]]) is called a Cauchy net if for every element $U\in \mathcal{U}$ there is an index $\alpha_0 \in A$ such that for all |
\[ | \[ | ||
(x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, . | (x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, . | ||
Latest revision as of 10:12, 9 December 2013
2020 Mathematics Subject Classification: Primary: 40A05 Secondary: 54E35 [MSN][ZBL]
Cauchy sequence, of points in a metric space $(X,d)$
A sequence $\{x_i\}$ of elements in a metric space $(X,d)$ such that for any $\varepsilon > 0$ there is a number $N$ such that \[ d (x_n, x_m) < \varepsilon \qquad \forall m,n\geq N\, . \] The latter is also called Cauchy condition. A convergent sequence is always necessarily a Cauchy sequence. However the converse is not necessarily true. A metric space with the property that any Cauchy sequence has a limit is called complete, see also Cauchy criteria.
The concept of Cauchy sequence can be generalized to Cauchy nets (see also Net; Net (of sets in a topological space), Generalized sequence and Cauchy filter) in a uniform space. Let $X$ be a uniform space with uniformity $\mathcal{U}$. A net $\{x_\alpha, \alpha \in A\}$ of elements $x_\alpha \in X$ (where $A$ is a directed set) is called a Cauchy net if for every element $U\in \mathcal{U}$ there is an index $\alpha_0 \in A$ such that for all \[ (x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, . \]
References
| [Al] | P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian) |
| [Du] | J. Dugundji, "Topology" , Allyn & Bacon (1966) MR0193606 Zbl 0144.21501 |
| [Ke] | J.L. Kelley, "General topology" , Springer (1975) |
| [KF] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) |
Cauchy sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_sequence&oldid=30874