Lemma 3.9.3.label Let $\pi:X\to Y$ be a surjective, continuous open map between topological spaces and $n\in\N^{+}$ then

1. (1)

   $\pi^{n}$ is an open surjetion, hence a quotient map

2. (2)

   For each topological space $Z$ and $f\in C(X^{n};Z)$ constant on $\pi$-fibers in each coordinate in the sense that for each $(x_{1},...,x_{n}).(x_{1}',...,x_{n}')\in X^{n}$

   \begin{align*}\pi(x_{i})=\pi(x_{i}'),\fall i=1,...,n\implies f(x_{1},...,x_{n})=f(x_{1}',...,x_{n}')\end{align*}

   
   

   then there uniquely exists $\til{f}\in C(Y^{n};Z)$ such that $\til{f}\circ\pi^{n}=f$.

3. (3)

   Let $\mu\in C(X^{n};X)$ such that $\mu$ is compatible with the equivalence relation induced by $\pi$ in the sense that for each $(x_{1},...,x_{n}).(x_{1}',...,x_{n}')\in X^{n}$

   \begin{align*}\pi(x_{i})=\pi(x_{i}'),\fall i=1,...,n\implies \pi(\mu(x_{1},...,x_{n}))=\pi(\mu(x_{1}',...,x_{n}'))\end{align*}

   
   

   then there uniquely exists $\til{\mu}\in C(Y^{n};Y)$ such that $\til{\mu}\circ\pi^{n}=\pi\circ \mu$.