Net (topology): Difference between revisions
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In [[topology]], a '''net''' is a [[function (mathematics)|function]] on a [[directed set]] into a [[topological space]] which generalises the notion of [[sequence]]. Convergence of a net may be used to completely characterise the topology. | In [[topology]], a '''net''' is a [[function (mathematics)|function]] on a [[directed set]] into a [[topological space]] which generalises the notion of [[sequence]]. Convergence of a net may be used to completely characterise the topology. | ||
A ''directed set'' is a [[partial order|partially ordered]] set ''D'' in which any two elements have a common upper bound. A ''net'' in a topological space ''X'' is a function ''a'' from a directed set ''D'' to ''X''. | A ''directed set'' is a [[partial order|partially ordered]] set ''D'' in which any two elements have a common upper bound. A ''net'' in a topological space ''X'' is a function ''a'' from a directed set ''D'' to ''X''. | ||
The [[natural number]]s with the usual order form a directed set, and so a [[sequence]] is a special case of a net. | |||
A net is ''eventually in'' a subset ''S'' of ''X'' if there is an index ''n'' in ''D'' such that for all ''m'' ≥ ''n'' we have ''a''(''m'') in ''S''. | A net is ''eventually in'' a subset ''S'' of ''X'' if there is an index ''n'' in ''D'' such that for all ''m'' ≥ ''n'' we have ''a''(''m'') in ''S''. | ||
Latest revision as of 17:12, 7 February 2009
In topology, a net is a function on a directed set into a topological space which generalises the notion of sequence. Convergence of a net may be used to completely characterise the topology.
A directed set is a partially ordered set D in which any two elements have a common upper bound. A net in a topological space X is a function a from a directed set D to X.
The natural numbers with the usual order form a directed set, and so a sequence is a special case of a net.
A net is eventually in a subset S of X if there is an index n in D such that for all m ≥ n we have a(m) in S.
A net converges to a point x in X if it is eventually in any neighbourhood of x.
References
- J.L. Kelley (1955). General topology. van Nostrand, 62-83.