A product space, also called a product topological space or direct product, is a Cartesian product equipped with the product topology.
For a family of topological spaces 
,
write 
and let 
be the coordinate map 
. The product
topology is the coarsest topology on 
making every 
a continuous map. A map 
from a topological space 
is continuous iff
every 
is continuous (Sharifi, p. 30). Equivalently,
any family of continuous
maps 
determines a unique continuous map 
satisfying 
for all 
, namely 
. This is the universal
property of the product space.
