Cauchy sequence
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=30877