Net (directed set)

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A mapping of a directed set into a (topological) space.


The topology of a space can be described completely in terms of convergence. However, this needs a more general concept of convergence than the concept of convergence of a sequence. What is needed is convergence of nets. A net $ S : D \rightarrow X $ in a topological space $ X $ converges to a point $ s \in X $ if for each open neighbourhood $ U $ of $ s $ in $ X $ the net $ S $ is eventually in $ U $. The last phrase means that there is an $ m \in D $ such that $ S ( n) \in U $ for all $ n \geq m $ in $ D $.

The theory of convergence of nets is known as Moore–Smith convergence, [a1].


[a1] J.L. Kelley, "General topology" , v. Nostrand (1955) pp. Chapt. II
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Net (directed set). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article