Definition 11

Let $f$ be an injective (resp. surjective) mapping of A into B. Any mapping $r$ (resp. $s$) of B into A such that $r \circ f$ (resp. $f \circ s$) is the identity mapping of A (resp. B) is called a retraction (resp. section) of $f$.

Instead of retraction (resp. section) the phrase left-inverse (resp. right-inverse) is sometimes used.

¶ If $f$ is injective (resp. surjective) and if $r$ (resp. $s$) is a retraction (resp. section) of $f$, then $f$ is a section (resp. retraction) of $r$ (resp. $s$). Hence a retraction is surjective and a section is injective.

¶ If $f$ is surjective and if $s$, $s'$ are two sections of $f$ such that $s(\mathrm{B}) = s'(\mathrm{B})$, then $s = s'$; for if $x \in \mathrm{B}$, there exists $y \in \mathrm{B}$ such that $s(x) = s'(y)$, and we have $x = f(s(x)) = f(s'(y)) = y$, so that $s(x) = s'(x)$ and consequently $s = s'$. Thus a section $s$ is uniquely determined by the set $s(\mathrm{B})$. By abuse of language, the set $s(\mathrm{B})$ is sometimes called a section of $f$.