Let be a set. A net is a map from a directed set to . In other words, it is a pair where is a directed set and is a map from to . If then is normally written , and then the net is written , or simply if the direct set is understood.
Nets are sometimes called Moore–Smith sequences, in which case convergence of nets may be called Moore–Smith convergence.
If is another directed set, and is an increasing map such that is cofinal in , then the pair is said to be a subnet of . Alternatively, a subnet of a net is sometimes defined to be a net such that for each there exists a such that for all .
Nets are a generalisation of sequences (http://planetmath.org/Sequence), and in many respects they work better in arbitrary topological spaces than sequences do. For example:
If is Hausdorff then any net in converges to at most one point.
If is a subspace of then if and only if there is a net in converging to .
if is another topological space and is a map, then is continuous at if and only if whenever is a net converging to , is a net converging to .
|Date of creation||2013-03-22 12:54:03|
|Last modified on||2013-03-22 12:54:03|
|Last modified by||yark (2760)|