Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
The identity map from $X$ to $X$ is trivially a clopen quotient map. – Brian M. Scott Jan 12 '13 at 18:42
And of course the constant map onto a single point. – Neal Jan 12 '13 at 19:32
up vote 3 down vote accepted

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).

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.