View source for Generalized sequence

''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.

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.

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



====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.