Let \(A\) and \(B\) be sets. A map \(\iota\colon A\to B\) is injective if for all \(x, y\in A\), \(\iota(x) = \iota(y)\) implies \(x = y\).
If \(A\neq\emptyset\), then \(\iota\colon A\to B\) is injective if and only if it has a left inverse: a map \(p\colon B\to A\) with \(p\circ\iota = \id_A\). The empty map \(\emptyset\to B\) with \(B\neq\emptyset\) is injective and has no left inverse. For a continuous injective map of topological spaces, a left inverse need not be continuous: the inclusion \(S^1\injects D^2\) has no continuous left inverse.
[@DF04].