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Difference between revisions of "Generalized sequence"

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''net''
 
''net''
  
A mapping of a [[Directed set|directed set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438701.png" /> into a (topological) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438702.png" />, i.e. a correspondence according to which each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438703.png" /> is associated with some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438704.png" />. A generalized sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438705.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438706.png" /> is convergent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438707.png" /> (sometimes one adds: with respect to the [[Directed order|directed order]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438708.png" />) to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g0438709.png" /> if for every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g04387010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g04387011.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g04387012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g04387013.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g04387014.png" />. This is the concept of Moore–Smith convergence [[#References|[3]]] (which seems more in conformity with intuitive ideas than convergence based on the concept of a [[Filter|filter]]). In terms of generalized sequences one can characterize the separation axioms (cf. [[Separation axiom|Separation axiom]]), various types of [[Compactness|compactness]] properties, as well as various constructions such as [[Compactification|compactification]], etc.
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A mapping of a [[Directed set|directed set]] $A$ into a (topological) space $X$, i.e. a correspondence according to which each $\alpha\in A$ is associated with some $x_\alpha\in X$. A generalized sequence $\{x_\alpha\colon\alpha\in A,\leq\}$ in a topological space $X$ is convergent in $X$ (sometimes one adds: with respect to the [[Directed order|directed order]] $\leq$) to a point $x\in X$ if for every neighbourhood $O_x$ of $x$ there exists a $\beta\in A$ such that $x_\alpha\in O_x$ for $\beta\leq\alpha\in A$. This is the concept of Moore–Smith convergence [[#References|[3]]] (which seems more in conformity with intuitive ideas than convergence based on the concept of a [[Filter|filter]]). In terms of generalized sequences one can characterize the separation axioms (cf. [[Separation axiom|Separation axiom]]), various types of [[Compactness|compactness]] properties, as well as various constructions such as [[Compactification|compactification]], etc.
  
 
Ordinary sequences constitute a special case of generalized sequences, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g04387015.png" /> is the set of natural numbers.
 
Ordinary sequences constitute a special case of generalized sequences, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043870/g04387015.png" /> is the set of natural numbers.
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====Comments====
 
====Comments====
The phrase  "generalized sequence"  is hardly ever used in the West; the commonly used terminology being  "net" . See also [[Net (directed set)|Net (directed set)]]. It should be noted that nets are necessary in the sense that sequences do not always suffice to characterize the various topological properties listed above.
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The phrase  "generalized sequence"  is hardly ever used in the West; the commonly used terminology being  "net". See also [[Net (directed set)|Net (directed set)]]. It should be noted that nets are necessary in the sense that sequences do not always suffice to characterize the various topological properties listed above.

Revision as of 19:05, 14 August 2014

net

A mapping of a directed set $A$ into a (topological) space $X$, i.e. a correspondence according to which each $\alpha\in A$ is associated with some $x_\alpha\in X$. A generalized sequence $\{x_\alpha\colon\alpha\in A,\leq\}$ in a topological space $X$ is convergent in $X$ (sometimes one adds: with respect to the directed order $\leq$) to a point $x\in X$ if for every neighbourhood $O_x$ of $x$ there exists a $\beta\in A$ such that $x_\alpha\in O_x$ for $\beta\leq\alpha\in A$. This is the concept of Moore–Smith convergence [3] (which seems more in conformity with intuitive ideas than convergence based on the concept of a filter). In terms of generalized sequences one can characterize the separation axioms (cf. Separation axiom), various types of compactness properties, as well as various constructions such as compactification, etc.

Ordinary sequences constitute a special case of generalized sequences, in which is the set of natural numbers.

References

[1] J.L. Kelley, "General topology" , v. Nostrand (1955)
[2] M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972)
[3] E.H. Moore, H.L. Smith, "A general theory of limits" Amer. J. Math. , 44 (1922) pp. 102–121


Comments

The phrase "generalized sequence" is hardly ever used in the West; the commonly used terminology being "net". See also Net (directed set). It should be noted that nets are necessary in the sense that sequences do not always suffice to characterize the various topological properties listed above.

How to Cite This Entry:
Generalized sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_sequence&oldid=18947
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article