net (original) (raw)

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 a∈A then γ⁢(a) is normally written xa, and then the net is written (xa)a∈A, or simply (xa) if the direct set A is understood.

Now suppose X is a topological space, A is a directed set, and (xa)a∈A is a net. Let x∈X. Then (xa) is said to converge to x if whenever U is an open neighbourhood of x, there is some b∈A such that xa∈U whenever a≥b.

Similarly, x is said to be an accumulation point (or cluster point) of (xa) if whenever U is an open neighbourhood of x and b∈Athere is a∈A such that a≥b and xa∈U.

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:B→A is an increasing map such that δ⁢(B) is cofinal in A, then the pair (B,γ∘δ)is said to be a subnet of (A,γ). Alternatively, a subnet of a net (xα)α∈Ais sometimes defined to be a net (xαβ)β∈Bsuch that for each α0∈Athere exists a β0∈Bsuch that αβ≥α0 for all β≥β0.

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:

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  If X is Hausdorff then any net in X converges to at most one point.
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  If Y is a subspace of Xthen x∈Y¯ if and only if there is a net in Y converging to x.
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  if X′ is another topological space and f:X→X′ 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).
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