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Net (directed set)

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


Comments

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 in a topological space converges to a point if for each open neighbourhood of in the net is eventually in . The last phrase means that there is an such that for all in .

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

References

[a1] J.L. Kelley, "General topology" , v. Nostrand (1955) pp. Chapt. II
How to Cite This Entry:
Net (directed set). M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Net_(directed_set)&oldid=17428
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098