Basic Analysis – Gently Done Topological Vector Spaces

2: Nets 19
A subset K of a metric space is compact if and only if any sequence in K has
a subsequence which converges to an element of K, but this need no longer be
true in a topological space. We will see an example of this later. We have seen
that in a general topological space nets can be used rather than sequences, so the
natural question is whether there is a sensible notion of that of ‘subnet’ of a net,
generalising that of subsequence of a sequence. Now, a subsequence of a sequence
is obtained simply by leaving out various terms—the sequence is labelled by the
natural numbers and the subsequence is labelled by a subset of these. The notion
of a subnet is somewhat more subtle than this.
Definition 2.15 A map F : J → I between directed sets J and I is said to be
cofinal if for any α ∈ I there is some β ′ ∈ J such that J(β) � α whenever β � β ′ .
In other words, F is eventually greater than any given α ∈ I.
Suppose that (x α ) α∈I is a net indexed by I and that F : J → I is a cofinal map
from the directed set J into I. The net (y β ) β∈J = (x F(β) ) β∈J is said to be a
subnet of the net (x α ) α∈I .
It is important to notice that there is no requirement that the index set for
the subnet be the same as that of the original net.
Example 2.16 If we set I = J = N, equipped with the usual ordering, and let
F : J → I be any increasing map, then the subnet (y n ) = (x F(n) ) is a subsequence
of the sequence (x n ).
Example 2.17 Let I = N with the usual order, and let J = N equipped with
the usual ordering on the even and odd elements separately but where any even
number is declared to be greater than any odd number. Thus I and J are directed
sets. Define F : J → I by F(β) = 3β. Let α ∈ I be given. Set β ′ = 2α so
that if β � β ′ in J, we must have that β is even and greater than β ′ in the usual
sense. Hence F(β) = 3β ≥ β ≥ β ′ = 2α ≥ α in I and so F is cofinal. Let
(x n ) n∈I be any sequence of real numbers, say. Then (x F(m) ) m∈J = (x 3m ) m∈J is a
subnet of (x n ) n∈I . It is not a subsequence because the ordering of the index set
is not the usual one. Suppose that x 2k = 0 and x 2k−1 = 2k − 1 for k ∈ I = N.
Then (x n ) is the sequence (1,0,3,0,5,0,7,0...). The subsequence (x 3m ) m∈N is
(3,0,9,0,15,0,...) which clearly does not converge in R. However, the subnet
(x 3m ) m∈J does converge, to 0. Indeed, for m � 2 in J, we have x 3m = 0.
Example 2.18 Let J = R, and let I = N, both equipped with their usual ordering,
and let F : R → N be any function such that F(t) → ∞ as t → ∞. Then F is
cofinal in N and (x F(t) ) t∈R is a subnet of the sequence (x n ) n∈N . This provides a
simple example of a subnet of a sequence which is not a subsequence.
We need to introduce a little more terminology.
Definition 2.19 The net (x α ) α∈I is said to be frequently in the set A if, for any
given γ ∈ I, x α ∈ A for some α ∈ I with α � γ.