Clopen quotient maps, does anyone know an example?

Question

A quotient map is a surjection $\mathbb{f} : X \rightarrow Y$ such that; $O$ is open in Y $\Leftrightarrow$ $\mathbb{f}^{-1}(O)$ is open in X.

This is an excercise in one of the later chapters of some book on topology basics. I'm quite inexperienced in constructing examples. So any help would be appreciated!

Answer 1

Let $H$ be a closed subgroup of a topological group $G$, where $G$ is assumed locally compact and Hausdorff. Equip $G/H$ with the quotient topology. Then the canonical quotient map $\pi:G\rightarrow G/H$ is clopen. You can find the details in any standard book on Harmonic analysis. I recommend the book "Principles of harmonic analysis" by Ditmar and Echterhoff (Lemma 1.1.5).

Answer 2

Continuous maps from compact Hausdorff spaces to Hausdorff spaces are always closed. (why?)

On the other hand, quotient maps coming from group actions are always open.

Thus one may cook up a large class of examples from a group acting properly discontinuously on compact Hausdorff spaces. (Properly discontinuous actions are one kind of actions that would make the quotient Hausdorff)

One example would be $S^n \to \mathbb{R}\mathbb{P}^n$ by identifying the antipodal points.