Namespaces
Variants
Actions

Difference between revisions of "Net (directed set)"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A mapping of a [[Directed set|directed set]] into a (topological) space.
+
<!--
 +
n0663301.png
 +
$#A+1 = 12 n = 0
 +
$#C+1 = 12 : ~/encyclopedia/old_files/data/N066/N.0606330 Net (directed set)
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
 +
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663301.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663302.png" /> converges to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663303.png" /> if for each open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663304.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663305.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663306.png" /> the net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663307.png" /> is eventually in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663308.png" />. The last phrase means that there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n0663309.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n06633010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n06633011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066330/n06633012.png" />.
+
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
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article