# Generalized sequence

2010 Mathematics Subject Classification: Primary: 54A20 [MSN][ZBL]

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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