Generalized sequence

2020 Mathematics Subject Classification: Primary: 54A20 [MSN][ZBL]

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 $(A,{\leq})$ is the set of natural numbers with the usual order.

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.

References

[a1]	L.A. Steen, J.A. Seebach Jr., "Counterexamples in topology", 2nd ed., Springer (1978) ISBN 0-387-90312-7 Zbl 0386.54001