The topology on the Cartesian product 
of two topological spaces whose open sets are
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 
.
In the definition of product topology of 
, where 
is any set, the open sets are the unions of subsets 
, where 
is an open subset of 
with the additional condition that 
for all but finitely many indices 
(this is automatically fulfilled if 
is a finite set). The reason for this choice of open sets
is that these are the least needed to make the projection onto the 
th factor 
continuous for all indices 
. Admitting all products of open sets would give rise to a
larger topology (strictly larger if 
is infinite), called the box topology.
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.
