= Projection map
A <projection map> from a <Cartesian product> selects specified coordinates. For the coordinate indexed by $j$,
$$
p_j:\prod_iX_i\longrightarrow X_j,\qquad p_j((x_i)_i)=x_j.
$$
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>.
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