Net (topology): Difference between revisions
Jump to navigation
Jump to search
imported>Bruce M. Tindall mNo edit summary |
mNo edit summary |
||
| Line 11: | Line 11: | ||
==References== | ==References== | ||
* {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=62-83 }} | * {{cite book | author=J.L. Kelley | authorlink=John L. Kelley | title=General topology | publisher=van Nostrand | year= 1955 | pages=62-83 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:01, 24 September 2024
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.