# Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function whose domain is the natural numbers. The codomain of this function is usually some topological space.

The motivation for generalizing the notion of a sequence is that, in the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map f between topological spaces X and Y:

1. The map f is continuous in the topological sense;
2. Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f(x) (continuous in the sequential sense).

While it is necessarily true that condition 1 implies condition 2, the reverse implication is not necessarily true if the toplogical spaces are not both first-countable. In particular, the two conditions are equivalent for metric spaces.

The concept of a net, first introduced by E. H. Moore and Herman L. Smith in 1922,[1] is to generalize the notion of a sequence so that the above conditions (with "sequence" being replaced by "net" in condition 2) are in fact equivalent for all maps of topological spaces. In particular, rather than being defined on a countable linearly ordered set, a net is defined on an arbitrary directed set. This allows for theorems similar to the assertion that the conditions 1 and 2 above are equivalent to hold in the context of topological spaces that do not necessarily have a countable or linearly ordered neighbourhood basis around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like directed sets in behaviour. The term "net" was coined by John L. Kelley.[2][3]

Nets are one of the many tools used in topology to generalize certain concepts that may only be general enough in the context of metric spaces. A related notion, that of the filter, was developed in 1937 by Henri Cartan.

## Definition

Let A be a directed set with preorder relation and X be a topological space with topology T. A function f: A → X is said to be a net.

If A is a directed set, we often write a net from A to X in the form (xα), which expresses the fact that the element α in A is mapped to the element xα in X.

A subnet is not merely the restriction of a net f to a directed subset of A; see the linked page for a definition.

## Examples of nets

Every non-empty totally ordered set is directed. Therefore, every function on such a set is a net. In particular, the natural numbers with the usual order form such a set, and a sequence is a function on the natural numbers, so every sequence is a net.

Another important example is as follows. Given a point x in a topological space, let Nx denote the set of all neighbourhoods containing x. Then Nx is a directed set, where the direction is given by reverse inclusion, so that ST if and only if S is contained in T. For S in Nx, let xS be a point in S. Then (xS) is a net. As S increases with respect to ≥, the points xS in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that xS must tend towards x in some sense. We can make this limiting concept precise.

## Limits of nets

If x = (xα)α ∈ A is a net from a directed set A into X, and if S is a subset of X, then we say that x is eventually in S (or residually in S) if there exists some α ∈ A such that for every β ∈ A with β ≥ α, the point xβ lies in S.

If x = (xα)α ∈ A is a net in the topological space X and xX then we say that the net converges to/towards x, that it has limit x, we call x a limit (point) of x, and write

xx        or        xαx        or        lim xx        or        lim xαx

if (and only if)

for every neighborhood U of x, x is eventually in U.

If lim xx and if this limit x is unique (uniqueness means that if lim xy then necessarily x = y) then this fact may be indicated by writing

lim x = x        or        lim xα = x

instead of lim xx.[4] In a Hausdorff space, every net has at most one limit so the limit of a convergent net in a Hausdorff space is always unique.[4] Some authors instead use the notation " lim x = x " to mean lim xx without also requiring that the limit be unique; however, if this notation is defined in this way then the equals sign = is no longer guaranteed to denote a transitive relationship and so no longer denotes equality (e.g. if x, yX are distinct and also both limits of x then despite lim x = x and lim x = y being written with the equals sign =, it is not true that x = y).

Intuitively, convergence of this net means that the values xα come and stay as close as we want to x for large enough α. The example net given above on the neighborhood system of a point x does indeed converge to x according to this definition.

Given a subbase B for the topology on X (where note that every base for a topology is also a subbase) and given a point xX, a net (xα) in X converges to x if and only if it is eventually in every neighborhood UB of x. This characterization extends to neighborhood subbases (and so also neighborhood bases) of the given point x.

## Supplementary definitions

Let φ be a net on X based on the directed set D and let A be a subset of X, then φ is said to be frequently in (or cofinally in) A if for every α in D there exists some β ≥ α, β in D, so that φ(β) is in A.

A point x in X is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood U of x, the net is frequently in U.

A net φ on set X is called universal, or an ultranet if for every subset A of X, either φ is eventually in A or φ is eventually in X − A.

## Examples

### Sequence in a topological space

A sequence (a1, a2, ...) in a topological space V can be considered a net in V defined on N.

The net is eventually in a subset Y of V if there exists an N in N such that for every nN, the point an is in Y.

We have limn anL if and only if for every neighborhood Y of L, the net is eventually in Y.

The net is frequently in a subset Y of V if and only if for every N in N there exists some nN such that an is in Y, that is, if and only if infinitely many elements of the sequence are in Y. Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.

### Function from a metric space to a topological space

Consider a function from a metric space M to a topological space V, and a point c of M. We direct the set M\{c} reversely according to distance from c, that is, the relation is "has at least the same distance to c as", so that "large enough" with respect to the relation means "close enough to c". The function f is a net in V defined on M\{c}.

The net f is eventually in a subset Y of V if there exists an a in M \ {c} such that for every x in M \ {c} with d(x,c) ≤ d(a,c), the point f(x) is in Y.

We have ${\displaystyle \lim _{x\to c}f(x)\to L}$ if and only if for every neighborhood Y of L, f is eventually in Y.

The net f is frequently in a subset Y of V if and only if for every a in M \ {c} there exists some x in M \ {c} with d(x,c) ≤ d(a,c) such that f(x) is in Y.

A point y in V is a cluster point of the net f if and only if for every neighborhood Y of y, the net is frequently in Y.

### Function from a well-ordered set to a topological space

Consider a well-ordered set [0, c] with limit point c, and a function f from [0, c) to a topological space V. This function is a net on [0, c).

It is eventually in a subset Y of V if there exists an a in [0, c) such that for every x ≥ a, the point f(x) is in Y.

We have ${\displaystyle \lim _{x\to c}f(x)\to L}$ if and only if for every neighborhood Y of L, f is eventually in Y.

The net f is frequently in a subset Y of V if and only if for every a in [0, c) there exists some x in [a, c) such that f(x) is in Y.

A point y in V is a cluster point of the net f if and only if for every neighborhood Y of y, the net is frequently in Y.

The first example is a special case of this with c = ω.

## Properties

Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence. The following set of theorems and lemmas help cement that similarity:

• A subset SX is open if and only if no net in XS converges to a point of S.[5] It is this characterization of open subsets that allows nets to characterize topologies.
• If U is a subset of X, then x is in the closure of U if and only if there exists a net (xα) with limit x and such that xα is in U for all α.
• A subset A of X is closed if and only if, whenever (xα) is a net with elements in A and limit x, then x is in A.
• A function f : XY between topological spaces is continuous at the point x if and only if for every net (xα) with
lim xαx
implies
lim f(xα) → f(x).
This theorem is in general not true if "net" is replaced by "sequence". We have to allow for directed sets other than just the natural numbers if X is not first-countable (or not sequential).
• In general, a net in a space X can have more than one limit, but if X is a Hausdorff space, the limit of a net, if it exists, is unique. Conversely, if X is not Hausdorff, then there exists a net on X with two distinct limits. Thus the uniqueness of the limit is equivalent to the Hausdorff condition on the space, and indeed this may be taken as the definition. This result depends on the directedness condition; a set indexed by a general preorder or partial order may have distinct limit points even in a Hausdorff space.
• The set of cluster points of a net is equal to the set of limits of its convergent subnets.
• A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet.
• A space X is compact if and only if every net (xα) in X has a subnet with a limit in X. This can be seen as a generalization of the Bolzano–Weierstrass theorem and Heine–Borel theorem.
• A net in the product space has a limit if and only if each projection has a limit. Symbolically, if (xα) is a net in the product X = πiXi, then it converges to x if and only if ${\displaystyle \pi _{i}(x_{\alpha })\to \pi _{i}(x)}$ for each i. Armed with this observation and the above characterization of compactness in terms on nets, one can give a slick proof of Tychonoff's theorem.
• If f : XY and (xα) is an ultranet on X, then (f(xα)) is an ultranet on Y.

## Cauchy nets

A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.[6]

A net (xα) is a Cauchy net if for every entourage V there exists γ such that for all α, β ≥ γ, (xα, xβ) is a member of V.[6][7] More generally, in a Cauchy space, a net (xα) is Cauchy if the filter generated by the net is a Cauchy filter.

## Relation to filters

A filter is another idea in topology that allows for a general definition for convergence in general topological spaces. The two ideas are equivalent in the sense that they give the same concept of convergence.[8] More specifically, for every filter base an associated net can be constructed, and convergence of the filter base implies convergence of the associated net—and the other way around (for every net there is a filter base, and convergence of the net implies convergence of the filter base).[9] For instance, any net ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ in ${\displaystyle X}$ induces a filter base of tails ${\displaystyle \{\{x_{\alpha }:\alpha \in A,\alpha _{0}\leq \alpha \}:\alpha _{0}\in A\}}$ where the filter in ${\displaystyle X}$ generated by this filter base is called the net's eventuality filter. This correspondence allows for any theorem that can be proven with one concept to be proven with the other.[9] For instance, continuity of a function from one topological space to the other can be characterized either by the convergence of a net in the domain implying the convergence of the corresponding net in the codomain, or by the same statement with filter bases.

Robert G. Bartle argues that despite their equivalence, it is useful to have both concepts.[9] He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common in analysis, while filters are most useful in algebraic topology. In any case, he shows how the two can be used in combination to prove various theorems in general topology.

## Limit superior

Limit superior and limit inferior of a net of real numbers can be defined in a similar manner as for sequences.[10][11][12] Some authors work even with more general structures than the real line, like complete lattices.[13]

For a net ${\displaystyle (x_{\alpha })_{\alpha \in I}}$ we put

${\displaystyle \limsup x_{\alpha }=\lim _{\alpha \in I}\sup _{\beta \succeq \alpha }x_{\beta }=\inf _{\alpha \in I}\sup _{\beta \succeq \alpha }x_{\beta }.}$

Limit superior of a net of real numbers has many properties analogous to the case of sequences, e.g.

${\displaystyle \limsup(x_{\alpha }+y_{\alpha })\leq \limsup x_{\alpha }+\limsup y_{\alpha },}$

where equality holds whenever one of the nets is convergent.

## Citations

1. ^ Moore, E. H.; Smith, H. L. (1922). "A General Theory of Limits". American Journal of Mathematics. 44 (2): 102–121. doi:10.2307/2370388. JSTOR 2370388.
2. ^ (Sundström 2010, p. 16n)
3. ^ Megginson, p. 143
4. ^ a b Kelley 1975, pp. 65-72.
5. ^ Howes 1995, pp. 83-92.
6. ^ a b Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, p. 260, ISBN 9780486131788.
7. ^ Joshi, K. D. (1983), Introduction to General Topology, New Age International, p. 356, ISBN 9780852264447.
8. ^ http://www.math.wichita.edu/~pparker/classes/handout/netfilt.pdf
9. ^ a b c R. G. Bartle, Nets and Filters In Topology, American Mathematical Monthly, Vol. 62, No. 8 (1955), pp. 551–557.
10. ^ Aliprantis-Border, p. 32
11. ^ Megginson, p. 217, p. 221, Exercises 2.53–2.55
12. ^ Beer, p. 2
13. ^ Schechter, Sections 7.43–7.47