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单词 Net
释义

net


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 ifwhenever 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 bAthere 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 mapsuch 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)αAis sometimes defined to be a net (xαβ)βBsuch that for each α0Athere exists a β0Bsuch that αβα0 for all ββ0.

Nets are a generalisation of sequences (http://planetmath.org/Sequence),and in many respects they work better in arbitrary topological spacesthan 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 Xthen xY¯ if and only if there is a net in Y converging to x.

  • if X is another topological spaceand f:XX is a map,then f is continuous at x if and only ifwhenever (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.

Titlenet
Canonical nameNet
Date of creation2013-03-22 12:54:03
Last modified on2013-03-22 12:54:03
Owneryark (2760)
Last modified byyark (2760)
Numerical id12
Authoryark (2760)
Entry typeDefinition
Classificationmsc 54A20
SynonymMoore-Smith sequence
Related topicFilter
Related topicNetsAndClosuresOfSubspaces
Related topicContinuityAndConvergentNets
Related topicCompactnessAndConvergentSubnets
Related topicAccumulationPointsAndConvergentSubnets
Related topicTestingForContinuityViaNets
Definessubnet
DefinesMoore-Smith convergence
Definescluster point
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更新时间:2025/5/5 0:06:57