Difference between revisions of "Net (directed set)"
From Encyclopedia of Mathematics
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| + | $#C+1 = 12 : ~/encyclopedia/old_files/data/N066/N.0606330 Net (directed set) | ||
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| + | A mapping of a [[Directed set|directed set]] into a (topological) space. | ||
====Comments==== | ====Comments==== | ||
| − | The topology of a space can be described completely in terms of convergence. However, this needs a more general concept of convergence than the concept of convergence of a sequence. What is needed is convergence of nets. A net | + | The topology of a space can be described completely in terms of convergence. However, this needs a more general concept of convergence than the concept of convergence of a sequence. What is needed is convergence of nets. A net $ S : D \rightarrow X $ |
| + | in a topological space $ X $ | ||
| + | converges to a point $ s \in X $ | ||
| + | if for each open neighbourhood $ U $ | ||
| + | of $ s $ | ||
| + | in $ X $ | ||
| + | the net $ S $ | ||
| + | is eventually in $ U $. | ||
| + | The last phrase means that there is an $ m \in D $ | ||
| + | such that $ S ( n) \in U $ | ||
| + | for all $ n \geq m $ | ||
| + | in $ D $. | ||
The theory of convergence of nets is known as Moore–Smith convergence, [[#References|[a1]]]. | The theory of convergence of nets is known as Moore–Smith convergence, [[#References|[a1]]]. | ||
Latest revision as of 08:02, 6 June 2020
A mapping of a directed set into a (topological) space.
Comments
The topology of a space can be described completely in terms of convergence. However, this needs a more general concept of convergence than the concept of convergence of a sequence. What is needed is convergence of nets. A net $ S : D \rightarrow X $ in a topological space $ X $ converges to a point $ s \in X $ if for each open neighbourhood $ U $ of $ s $ in $ X $ the net $ S $ is eventually in $ U $. The last phrase means that there is an $ m \in D $ such that $ S ( n) \in U $ for all $ n \geq m $ in $ D $.
The theory of convergence of nets is known as Moore–Smith convergence, [a1].
References
| [a1] | J.L. Kelley, "General topology" , v. Nostrand (1955) pp. Chapt. II |
How to Cite This Entry:
Net (directed set). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Net_(directed_set)&oldid=47955
Net (directed set). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Net_(directed_set)&oldid=47955
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article