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net∗
yark†
2013-03-21 14:35:41
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 ∈ A there 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∈A is sometimes defined to be a net (xαβ )β∈B
such that for each α0 ∈ A there exists a β0 ∈ B such that αβ ≥ α0 for all β ≥ β0 .
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 Hausdorff then any net in X converges to at most one point.
• If Y is a subspace of X then x ∈ Y if and only if there is a net in Y
converging to x.
• if X 0 is another topological space and f : X → X 0 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 convergent subnet.
∗ hNeti
created: h2013-03-21i by: hyarki version: h33250i Privacy setting: h1i
hDefinitioni h54A20i
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