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box topology∗
yark†
2013-03-21 14:22:22
Let {(Xα , Tα )}α∈A be a family of topological spaces. Let Y denote the
generalized Cartesian product of the sets Xα , that is
Y
Y =
Xα .
α∈A
Let B denote the set of all products of open sets of the corresponding spaces,
that is
(
)
Y
B=
Uα Uα ∈ Tα for all α ∈ A .
α∈A
Now we can construct the box product (Y, S), where S, referred to as the box
topology, is the topology generated by the base B.
When A is a finite set, the box topology coincides with the product topology.
Example
As an example, the box product of two topological spaces (X0 , T0 ) and (X1 , T1 )
is (X0 × X1 , S), where the box topology S
S (which is the same as the product
topology) consists of all sets of the form i∈I (Ui × Vi ), where I is some index
set and for each i ∈ I we have Ui ∈ T0 and Vi ∈ T1 .
∗ hBoxTopologyi
created: h2013-03-21i by: hyarki version: h33095i Privacy setting: h1i
hDefinitioni h54A99i
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