Basic Analysis – Gently Done Topological Vector Spaces

3. Product Spaces
Let(X 1 ,T 1 )and(X 2 ,T 2 )betopologicalspacesandletY betheircartesianproduct
Y = X 1 ×X 2 = {(x 1 ,x 2 ) : x 1 ∈ X 1 , x 2 ∈ X 2 }.
We have discussed the product topology on Y in example 2 of Examples 1.2 and in
Example 1.38. Those sets of the form U×V, with U ∈ T 1 and V ∈ T 2 , form a base
and, since U ×V = U ×X 2 ∩X 1 ×V, the sets {U×X 2 , X 1 ×V : U ∈ T 1 , V ∈ T 2 }
constitute a sub-base for the product topology.
The projection maps, p 1 and p 2 , on the cartesian product X 1 ×X 2 , are defined
by
p 1 : X 1 ×X 2 → X 1 , (x 1 ,x 2 ) ↦→ x 1
p 2 : X 1 ×X 2 → X 1 , (x 1 ,x 2 ) ↦→ x 2 .
Then the product topology is the weakest topology on the cartesian product X 1 ×
X 2 such that both p 1 and p 2 are continuous—the σ(X 1 ×X 2 ,{p 1 ,p 2 })-topology.
We would liketo generalise thisto an arbitrary cartesian product of topological
spaces. Let {(X α ,T α ) : α ∈ I} be a collection of topological spaces indexed by
the set I. We recall that X = �
αXα , the cartesian product of the Xα ’s, is
defined to be the collection of maps γ from I into the union �
α X α satisfying
γ(α) ∈ Xα for each α ∈ I. We can think of the value γ(α) as the α-coordinate of
the point γ in X. The idea is to construct a topology on X = �
αXα built from
the individual topologies T α . Two possibilities suggest themselves. The first is the
weakest topology on X with respect to which all the projection maps p α → X α
are continuous. The second is to construct the topology on X whose open sets
are unions of ‘super rectangles’, that is, sets of the form �
α U α , where U α ∈ T α
for every α ∈ I. In general, these two topologies are not the same, as we will see.
We take the first of these as our definition of the product topology for arbitrary
products.
Definition 3.1 The product topology, denoted Tprod on the cartesian product of
the topological spaces {(Xα ,Tα ) : α ∈ I} is the σ( �
αXα ,F)-topology, where F is
the family of projection maps {pα : α ∈ I}.
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