Basic Analysis – Gently Done Topological Vector Spaces

20 Basic Analysis
A point z is a cluster point of the net (x α ) α∈I if (x α ) α∈I is frequently in any
neighbourhood of z.
Note that if x α = x for every α ∈ I, then x is certainly a cluster point of the
net (x α ) α∈I . However, x is not a limit point of the point set in X determined by
the net, namely {x α : α ∈ I}, since this is just the one-point set {x}, which has
no limit points at all.
Proposition 2.20 Let (x α ) I be a net in the space X and let A be a family of
subsets of X such that
(i) (x α ) I is frequently in each member of A;
(ii) for any A,B ∈ A there is C ∈ A such that C ⊆ A∩B.
Then there is a subnet (x F(β) ) J of the net (x α ) I such that (x F(β) ) J is eventually
in each member of A.
Proof Equip A with the ordering given by reverse inclusion, that is, we define
A � B to mean B ⊆ A for A,B ∈ A. For any A,B ∈ A, there is C ∈ A with
C ⊆ A ∩ B, by (ii). This means that C � A and C � B and we see that A is
directed with respect to this partial ordering.
Let E denote the collection of pairs (α,A) ∈ I ×A such that x α ∈ A;
E = {(α,A) : α ∈ I, A ∈ A, x α ∈ A}.
Define (α ′ ,A ′ ) � (α ′′ ,A ′′ ) to mean that α ′ � α ′′ in I and A ′ � A ′′ in A. Then
� is a partial order on E. Furthermore, for given (α ′ ,A ′ ),(α ′′ ,A ′′ ) in E, there is
α ∈ I with α � α ′ and α � α ′′ , and there is A ∈ A such that A � A ′ and A � A ′′ .
But (x α ) is frequently in A, by (i), and therefore there is β � α ∈ I such that
x β ∈ A. Thus (β,A) ∈ E and (β,A) � (α,A ′ ), (β,A) � (α,A ′′ ) and it follows that
E is directed. E will be the index set for the subnet.
Next, we must construct a cofinal map from E to I. Define F : E → I by
F((α,A)) = α. To show that F is cofinal, let α 0 ∈ I be given. For any A ∈ A
there is α � α 0 such that x α ∈ A (since (x α ) is frequently in each A ∈ A. Hence
(α,A) ∈ E and F((α,A)) = α � α 0 . So if (α ′ ,A ′ ) � (α,A) in E, then we have
F((α ′ ,A ′ )) = α ′ � α � α 0 .
This shows that F is cofinal and therefore (x F((α,A)) ) E is a subnet of (x α ) I .
It remains to show that this subnet is eventually in every member of A. Let
A ∈ A be given. Then there is α ∈ I such that x α ∈ A and so (α,A) ∈ E. For any
(α ′ ,A ′ ) ∈ E with (α ′ ,A ′ ) � (α,A), we have
Thus (x F((α,A)) ) E is eventually in A.
x F((α ′ ,A ′ )) = x α ′ ∈ A ′ ⊆ A.
Department of Mathematics King’s College, London