A mapping of a directed set into a (topological) space , i.e. a correspondence according to which each is associated with some . A generalized sequence in a topological space is convergent in (sometimes one adds: with respect to the directed order ) to a point if for every neighbourhood of there exists a such that for . This is the concept of Moore–Smith convergence  (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.
|||J.L. Kelley, "General topology" , v. Nostrand (1955)|
|||M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972)|
|||E.H. Moore, H.L. Smith, "A general theory of limits" Amer. J. Math. , 44 (1922) pp. 102–121|
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.
Generalized sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_sequence&oldid=18947