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.

For the coordinate projection map $p_i:X_1\times X_2\to X_i$, let $X_j$ denote the other factor. Then $p_i$ is surjective exactly when $X_i$ is the empty set or $X_j$ is nonempty. If $X_i$ is nonempty, a preimage of one chosen element requires an element of $X_j$, and that same element provides a preimage for every element of $X_i$.

The projection map is injective exactly when $X_i$ is the empty set or $X_j$ has at most one element. When $X_i$ contains an element and $X_j$ contains two distinct elements, the corresponding distinct ordered pairs have the same projection. With at most one omitted coordinate available, two pairs with the same retained coordinate must agree.

These conditions include the cases with the empty set as the domain of a function. In particular, a surjection onto an empty Cartesian product need not induce a surjection onto a nonempty factor: the empty function onto $\{a\}\times \varnothing$ is surjective, while the resulting empty function into $\{a\}$ is not.