 Projection map

A projection map from a Cartesian product selects specified coordinates. For the coordinate indexed by $j$,For example, $p_1(a,b)=a$ forgets the second component of an ordered pair.

The product topology makes each coordinate projection map a continuous map. In the two-factor case $X\times Y\to X$, surjectivity depends on whether the omitted factor $Y$ has an element, subject to the empty-set cases for Cartesian projections. For an arbitrary infinite family, simultaneous existence of the other coordinates can also involve the axiom of choice.