Let X be a set. A net is a map from a directed set to X. In other words, it is a pair (A,γ) where A is a directed set and γ is a map from A to X. If aA then γ(a) is normally written xa, and then the net is written (xa)aA, or simply (xa) if the direct set A is understood.

Now suppose X is a topological spaceMathworldPlanetmath, A is a directed set, and (xa)aA is a net. Let xX. Then (xa) is said to converge to x if whenever U is an open neighbourhood of x, there is some bA such that xaU whenever ab.

Similarly, x is said to be an accumulation pointMathworldPlanetmathPlanetmath (or cluster point) of (xa) if whenever U is an open neighbourhood of x and bA there is aA such that ab and xaU.

Nets are sometimes called Moore–Smith sequences, in which case convergence of nets may be called Moore–Smith convergence.

If B is another directed set, and δ:BA is an increasing map such that δ(B) is cofinal in A, then the pair (B,γδ) is said to be a subnet of (A,γ). Alternatively, a subnet of a net (xα)αA is sometimes defined to be a net (xαβ)βB such that for each α0A there exists a β0B such that αβα0 for all ββ0.

Nets are a generalisation of sequences (, and in many respects they work better in arbitrary topological spaces than sequences do. For example:

  • If X is HausdorffPlanetmathPlanetmath then any net in X converges to at most one point.

  • If Y is a subspaceMathworldPlanetmath of X then xY¯ if and only if there is a net in Y converging to x.

  • if X is another topological space and f:XX is a map, then f is continuous at x if and only if whenever (xa) is a net converging to x, (f(xa)) is a net converging to f(x).

  • X is compact if and only if every net has a convergentMathworldPlanetmathPlanetmath subnet.

Title net
Canonical name Net
Date of creation 2013-03-22 12:54:03
Last modified on 2013-03-22 12:54:03
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Definition
Classification msc 54A20
Synonym Moore-Smith sequence
Related topic Filter
Related topic NetsAndClosuresOfSubspaces
Related topic ContinuityAndConvergentNets
Related topic CompactnessAndConvergentSubnets
Related topic AccumulationPointsAndConvergentSubnets
Related topic TestingForContinuityViaNets
Defines subnet
Defines Moore-Smith convergence
Defines cluster point