Net (mathematics)

:This article is about nets in topological spaces and not about ε-nets in metric spaces

Related Topics:
Topological space - ε-net - Metric space

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In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. Limits of nets accomplish for all topological spaces what limits of sequences accomplish for first-countable spaces such as metric spaces.

Related Topics:
Topology - Mathematics - Sequence - Limit - Topological space - First-countable space - Metric space

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A sequence is usually indexed by the natural numbers which are a totally ordered set. Nets generalize this concept by weakening the order relation on the index set to that of a directed set.

Related Topics:
Natural numbers - Totally ordered set - Order relation - Directed set

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Nets were first introduced by E. H. Moore and H. L. Smith in 1922. An equivalent notion, called filter, was developed in 1937 by Henri Cartan.

Related Topics:
E. H. Moore - H. L. Smith - 1922 - Filter - 1937 - Henri Cartan

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