Basic Analysis – Gently Done Topological Vector Spaces

3: Product Spaces 31
point of (γ α ↾ J) α∈A in �
j∈J X j , it is the limit of some subnet (γ φ(β) ↾ J) β∈B ,
say. Now, (γ φ(β) (k)) β∈B is a net in the compact space X k and therefore has a
cluster point, ξ ∈ X k , say. Let J ′ = J ∪{k} and define γ ′ ∈ �
j∈J ′ X j by
γ ′ �
γ(j), j ∈ J
(j) =
ξ, j = k.
We shall show that γ ′ is a cluster point of (γ α ↾ J ′ ) α∈A . Let F be any finite subset
in J and, for j ∈ F, let U j be any open neighbourhood of γ ′ (j) in X j , and let
V be any open neighbourhood of γ ′ (k) = ξ in X k . Since (γ φ(β) ) β∈B converges
to γ in �
j∈J Xj , there is β1 ∈ B such that γφ(β) (j) β∈B ∈ Uj for each j ∈ F for
all β � β1 . Furthermore, (γφ(β) (k) β∈B is frequently in V. Let α0 ∈ A be given.
There is β0 ∈ B such that if β � β0 then φ(β) � α0 . Let β2 ∈ B be such that
β2 � β0 and β2 � β1 . Since (γφ(β) (k) β∈B is frequently in V, there is β � β2 such
that γφ(β) (k) ∈ V. Set α = φ(β) ∈ A. Then α � α0 , γα (k) ∈ V and, for j ∈ F,
γα (j) = γφ(β) (j) ∈ Uj . It follows that γ ′ is a cluster point of the net (γα ↾ J ′ ) α∈A ,
as required. This means that γ ′ ∈ P. However, it is clear that γ � γ ′ and that
γ �= γ ′ . This contradicts the maximality of γ in P and we conclude that, in fact,
J = I and therefore γ is a cluster point of (γα ) α∈A .
We have seen that any net in X has a cluster point and therefore it follows
that X is compact.
Finally, we will consider a proof using universal nets.
Proof (version 3) Let (γα ) α∈A be any universal net in X = �
i∈I Xi . For any
i ∈ I, let Si be any given subset of Xi and let S be the subset of X given by
S = {γ ∈ X : γ(i) ∈ S i }.
Then (γ α ) is either eventually in S or eventually in X \ S. Hence we have that
either (γ α (i)) is either eventually in S i or eventually in X i \S i . In other words,
(γ α (i)) α∈A is a universal net in X i . Since X i is compact, by hypothesis, (γ α (i))
converges; γ α (i) → x i , say, for i ∈ I. Let γ ∈ X be given by γ(i) = x i , i ∈ I. Then
we have that p i (γ α ) = γ α (i) → x i = γ(i) for each i ∈ I and therefore γ α → γ in
X. Thus every universal net in X converges, and we conclude that X is compact.