Proposition 6

Let $X$ be a topological space, $R$ an equivalence relation on $X$, $\varphi$ the canonical mapping of $X$ onto $X/R$; then a mapping $f$ of $X/R$ into a topological space $Y$ is continuous if and only if $f \circ \varphi$ is continuous on $X$.

This is a particular case of § 2, no. 4, Proposition 6; it expresses the fact that the quotient topology is the final topology for the mapping $\varphi$.

Proposition 6 shows that there is a one-to-one correspondence between the continuous mappings of $X/R$ into $Y$ and the continuous maps of $X$ into $Y$ which are constant on each equivalence class mod $R$.