Different identifications of the same sides of a polygon make the same quotient space

Question

Let P$\subset\mathbb{R}^{2}$ be a polygon with sides $l_{1},...,l_{n}$ parametrized by the curves $\alpha_{1}(t),...,\alpha_{n}(t)$. Let $\beta_{1}(t),...,\beta_{n}(t)$ be another parametrization of the same sides, with $\beta_{i}(0)=\alpha_{i}(0)$, $\beta_{i}(1)=\alpha_{i}(1)$. Let $A_{1},...,A_{r}$ be a partition of $\{1,...,n\}$. Then $x\sim y$ if and only if there exists $t\in[0,1]$ so that $x=\alpha_{i}(t)$, $y=\alpha_{j}(t)$ for certain $i,j\in A_{s}, s\in\{1,...,r\}$ and $x\approx y$ if and only if there exists $t\in[0,1]$ so that $x=\beta_{i}(t)$, $y=\beta_{j}(t)$ for certain $i,j\in A_{m}, m\in\{1,...,r\} $. I have to prove that $P/\sim$ and $P/\approx$ are homeomorphic.

I think I have to find an homeomorphism $\Psi:P\rightarrow P$ that descends to the quotient, but I do not find it.

Answer 1

Let $P_i$ denote the $i^{th}$ side of $P$ and $\partial P$ denote the boundary of $P$.

Hints:

1. Show that for each $i$, there is a homeomorphism $f_i:P_i\to P_i$ such that $\beta_i = f_i\circ\alpha_i$.

2. Show this defines a homeomorphism $f:\partial P\to \partial P$.

3. Extend $f$ to $\tilde{f}: P\to P$.

4. Show that $\tilde{f}$ is a homeomorphism and it descends to the quotient.