Nets, also known as generalised sequences, extend the concept of a sequence in that they allow to apply most of the present chapter’s results to spaces that may not be first countable.
Definition 6.55 An ordered set(I, ≤) is called directed when, for every i, j ∈ I ,
there exists h∈ I such that h ≥ i and h ≥ j.
Equivalently, a directed set is an ordered set in which every finite subset is upper bounded. Let us present a few examples:
1. the setN with the usual ordering and, more generally, Nnwith the order relation
(a1, . . . , an) ≤ (b1, . . . , bn) ⇐⇒ ai ≤ bi for every i ;
2. the setP0(A) of finite subsets of A with the inclusion;
3. the set I(x) of neighbourhoods of a point x in a topological space, with the relation U ≤ V ⇐⇒ V ⊂ U.
126 6 Sequences Definition 6.56 A map p: J → I between directed sets is called a cofinal morphism if it preserves orderings ( p( j) ≥ p( j) if j ≥ j) and for every i ∈ I there’s a j ∈ J such that p( j) ≥ i.
Example 6.57 Consider the family I of finite subsets inside a set A with even car-
dinality. The inclusion I → P0(A) is a cofinal morphism.
The composite of two cofinal morphisms is a cofinal morphism.
Definition 6.58 A net in a topological space X is a map f: I → X, where I is
directed.
For I = N, a net f : I → X is nothing but a sequence in X. The name ‘net’ is inspired by the case I = Nn.
Definition 6.59 Let X be topological space, f: I → X a net and x a point in X.
One says that:
1. the net f converges to x if for any neighbourhood U ∈ I(x) there’s an index
i∈ I such that f ( j) ∈ U for every j ≥ i;
2. x is a limit point of the net f if for any neighbourhood U∈ I(x) and any i ∈ I there’s a j≥ i such that f ( j) ∈ U.
In a similar manner to what occurs with sequences, a converging net must converge to a limit point, whilst a limit point isn’t necessarily a point to which the net converges, in general.
Lemma 6.60 Let f: I → X be a net. Then x ∈ X is a limit point for f if and only
if there’s a cofinal morphism p: J → I such that the net f p : J → X converges to x.
Proof If f p converges to some x, then x is a limit point of f because p is cofinal.
As for the converse: suppose x is a limit point of f and callI(x) the family of neighbourhoods around x, as usual. Consider
J= {(i, U) ∈ I × I(x) | f (i) ∈ U},
with the relation
(i, U) ≤ ( j, V ) if i ≤ j and V ⊂ U.
The set(J, ≤) is directed: if (i, U), ( j, V ) ∈ J, there’s an h ∈ I such that h ≥ i and h≥ j. By definition of limit point, there exists k ≥ h such that f (k) ∈ U ∩ V , so(k, U ∩ V ) ≥ (i, U), (k, U ∩ V ) ≥ ( j, V ). The projection onto the first factor
p: J → I is surjective, hence a cofinal morphism (just notice that p(i, X) = i for
every i ∈ I ). Let U be a neighbourhood of x and take i ∈ I such that f (i) ∈ U. Then for every( j, V ) ≥ (i, U) we have f p( j, V ) ∈ U, and f p converges to x.
Proposition 6.61 Let A be a subset in a space X . For any point x ∈ X the following
are equivalent requirements:
1. there is a net in A that converges to x; 2. there is a net in A with limit point x; 3. the point x belongs in the closure of A.
Proof (1) ⇒ (2) is trivial. If x ∈ X is a limit point of some net f : I → A, for any
neighbourhood U of x there’s an i ∈ I such that f (i) ∈ U; hence U ∩ f (I ) = ∅, i.e.
x belongs to the closure of f(I ). As f (I ) ⊂ A we have x ∈ A, proving (2) ⇒ (3).
Let us see that(3) ⇒ (1). If x ∈ A and I(x) is the directed set of neighbourhoods around x, the axiom of choice guarantees that we may pick a net f: I(x) → A such that f(U) ∈ U ∩ A for every neighbourhood U of x. Immediately, then, f converges
to x.
Theorem 6.62 A topological space X is compact if and only if every net has a limit
point.
Proof Suppose X is compact and there’s a net f : I → X without limit points:
this will lead to a contradiction. For any x ∈ X (not a limit point) there are a neighbourhood U(x) ∈ I(x) and an index i(x) ∈ I such that f ( j) ∈ U(x) for any
j ≥ i(x). By compactness we have x1, . . . , xn ∈ X such that X = U(x1) ∪ · · · ∪
U(xn). Let h ∈ I be an upper bound for i(x1), . . . , i(xn). Then f (h) ∈ U(xi) for
any i = 1, . . . , n, absurd.
Vice versa, assume X isn’t compact and choose an open cover X = ∪{Ua| a ∈ A}
without finite subcovers. Consider the directed setP0(A) of finite subsets of A. Using
the axiom of choice we pick a net f: P0(A) → X such that f (B) ∈ Uafor any
B∈ P0(A) and every a ∈ B. If x ∈ Ua⊂ X were a limit point, there would exist a
finite subset B ⊂ A containing a and such that f (B) ∈ Ua. But this cannot be, so
the net f can’t have limit points.
Example 6.63 To get an idea of but one possible application of nets, we outline
the proof of the fact that every finite-dimensional subspace of a topological vector
space is closed. The reader will find extremely worthwhile filling in all the gaps. For
simplicity let’s consider real vector spaces: for complex vectors spaces the argument is completely analogous.
A topological vector space V is a vector space furnished with a Hausdorff topol- ogy that makes the operation of taking linear combinations continuous; in particu- lar, for any linearly independent vectorsv1, . . . , vn the linear map f: Rn → V ,
f(x1, . . . , xn) =
xivi is continuous and 1-1. Let us prove f is a closed immersion. Take C ⊂ Rn closed and v ∈ f (C); we have to show v ∈ f (C). Let s: I → f (C) be a net converging to v ∈ V and write t = f−1◦ s : I → C. Next, immerseRnin its one-point compactificationRn∪{∞} Sn; we may assume that limit(i) ∈ Rn∪{∞} exists, possibly composing with another cofinal morphism.
There are two cases to consider, in the first of which the limit of t is contained in Rn; as C is closed inRnwe havev = f (limit(i)) ∈ f (C).
128 6 Sequences
In the second case limit(i) = ∞, so limit(i) = +∞ and limi s(i)
t(i) = 0. On the other hand s(i)
t(i) = f
t(i)
t(i)
, and the net i → t(i)
t(i) has at least one limit point a∈ Sn−1. Therefore f(a) = 0, contradicting the injectivity of f .
Exercises
6.34 Prove that a space is Hausdorff if and only if every net converges to one point,
at most.
6.35 Say if the following nets a: N × N → R converge, and if so compute the
limits:
a(n, m) = 1
n+ m, a(n, m) =
1
(n − m)2+ 1, a(n, m) = ne−m. 6.36 Let S be a non-empty set and a: S → R any function. Explain the meaning of
the expression ‘the seriess∈S f(s) converges’. (Hint: consider the directed set of
finite subsets of S.)
6.37 (K) Let {an} be a real sequence. Consider the directed set P0(N) of finite
subsets inN and the net
f: P0(N) → R, f(A) =
n∈A an.