The product topology on the Cartesian product 
of two topological
spaces has as its open sets the unions
of subsets 
, where 
and 
are open subsets of 
and 
, respectively.
This definition extends in a natural way to the Cartesian product of any finite number 
of topological spaces.
The product topology of
 |
where 
is the real line with the Euclidean
topology, coincides with the Euclidean topology
of the Euclidean space 
.
For 
, where 
is any set, the product topology has
as its open sets the unions
of subsets 
, where 
is an open
subset of 
and 
for all but finitely many indices 
. This condition is automatic when 
is finite. The product topology
is the coarsest topology making every coordinate map 
, defined by 
, continuous
(Sharifi, p. 30).
Allowing all products of open sets gives the box topology, which contains the product topology. The two agree for finite
products and may agree even when 
is infinite: a product of singleton sets has only a single point
and a single topology. In contrast, 
is open in the box
topology on 
but not in its product topology. Every basic product neighborhood
of the origin leaves some coordinate unrestricted and therefore contains a point
outside 
(Sharifi, p. 32).
The product topology is also called Tychonoff topology, but this should not cause any confusion with the notion of Tychonoff space, which has a completely different meaning.
