Lemma 1

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a continuous mapping. The following conditions are equivalent:

(i) For every topological space Z and every continuous mapping $g: X\rightarrow Z$, there exists a continuous mapping $g': Y\rightarrow Z$ such that $g=$ $g'\circ f$;

(ii) The mapping $f$ is injective and possesses a continuous retraction ;

(iii) The mapping $f$ defines a homeomorphism of X onto its image $f(X)$, which is a retract of Y.

Suppose that $f$ is injective and that $r: Y\rightarrow$ X is a continuous retraction of the mapping $f$. For every continuous mapping $g: X\rightarrow Z$, the mapping $g'=g\circ r: Y\rightarrow Z$ satisfies $g'\circ f=g\circ r\circ f=g$. This proves that (ii)$\Rightarrow$(i).

Suppose that condition (i) is satisfied. Let $g: X\rightarrow X$ be the identity mapping. By hypothesis, there exists a continuous mapping $g': Y\rightarrow X$ such that $g'\circ f=g$. This implies that the mapping $f$ is injective and that $g'$ is a continuous retraction of $f$.

The equivalence of the properties (ii) and (iii) is immediate.