Show that a Quotient Map pasting a polygonal region is closed

Question

Given a polygonal region $P$, a labeling equivalence relation and a pasting quotient map $\pi: P \to X$, show that $\pi$ is closed.

For any set $C$ closed in $P$, show that $\pi(C)$ is closed.

I'm struggling with how to approach this one.

Answer 1

We know that $P$ is comopact. To show the image $\pi(C)$ is closed directly may not be easy, but we can show $\pi^{-1}(\pi(C))$ is closed. claim:

1. $\pi^{-1}(\pi(C))$ is the set of all points that are identified with $C$.

2. There must exist a set of edges $\{e_i|i\in I\}$ (if this set is empty then we're done because all points but edges are equivalent to only themselves) for some index set $I$ that are mapped to $C$ by a family of homeomorphisms $h_i:e_i\to e$, where $e$ is an edge.

So, we have $\pi^{-1}(\pi(C))=\bigcup_{i\in I}h_i(e_i)\cup (C\cap e)\cup C$ which is compact and, therefore, closed in $P$. And because of the surjectivity, we have $\pi(\pi^{-1}(\pi(C)))=\pi(C)$ (surjections are right invertible), which is also compact (by continuity) and hence closed as desired.