Net (mathematics) - Properties

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 function ƒ:X→ Y between topological spaces is continuous at the point x if and only if for every net (x_α) with

lim x_α = x

we have

lim ƒ(x_α) = ƒ(x).

Note that this theorem is in general not true if we replace "net" by "sequence". We have to allow for more directed sets than just the natural numbers if X is not first-countable.

• 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. Note that 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.

• If U is a subset of X, then x is in the closure of U if and only if there exists a net (x_α) with limit x and such that x_α is in U for all α.

• A subset A of X is closed if and only if, whenever (x_α) is a net with elements in A and limit x, then x is in A.

• The set of cluster points of a net is equal to the set of limits of its convergent subnets.

Proof

Let X be a topological space, A a directed set, be a net in X, and . It is easily seen that if y is a limit of a subnet of, then y is a cluster point of . Conversely, assume that y is a cluster point of . Let B be the set of pairs where U is an open neighborhood of y in X and is such that . The map mapping to is then cofinal. Moreover, giving B the product order (the neighborhoods of y are ordered by inclusion) makes it a directed set, and the net defined by converges to y.

• 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 (x_α) 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.

Proof

First, suppose that X is compact. We will need the following observation (see Finite intersection property). Let I be any set and be a collection of closed subsets of X such that for each finite . Then as well. Otherwise, would be an open cover for X with no finite subcover contrary to the compactness of X. Let A be a directed set and be a net in X. For every define The collection has the property that every finite subcollection has non-empty intersection. Thus, by the remark above, we have that and this is precisely the set of cluster points of . By the above property, it is equal to the set of limits of convergent subnets of . Thus has a convergent subnet. Conversely, suppose that every net in X has a convergent subnet. For the sake of contradiction, let be an open cover of X with no finite subcover. Consider . Observe that D is a directed set under inclusion and for each, there exists an such that for all . Consider the net . This net cannot have a convergent subnet, because for each there exists such that is a neighbourhood of x; however, for all, we have that . This is a contradiction and completes the proof.

• A net in the product space has a limit if and only if each projection has a limit. Symbolically, if (x_α) is a net in the product X = π_iX_i, then it converges to x if and only if 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 ƒ:X→ Y and (x_α) is an ultranet on X, then (ƒ(x_α)) is an ultranet on Y.