The box topology on a Cartesian product 
of topological
spaces is the topology having as a topological
basis all sets of the form
 |
where each 
is an open set in 
(Sharifi, p. 32). The product
topology uses only such products with 
for all but finitely
many 
. Thus every open
set in the product topology is open
in the box topology. The two topologies coincide when
is finite
(Sharifi, pp. 30 and 32).
When 
is infinite,
the box topology can be strictly finer. For example, on 
with the usual Euclidean
topology in each factor, let 
index the positive integers.
The set 
is open in the box topology
but not in the product topology. Every basic
open set in the product
topology containing the zero sequence leaves all
but finitely many factors unrestricted, so it contains
a point outside 
(Sharifi, pp. 32-33).
