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

Consider the non-negative real line with identification $x\sim 2x$. Is there a special name for this quotient space; what's known about it?

share|cite|improve this question
Without the point $\{0\}$ this space would just be a circle (think of the homeomorphism $\log\colon\mathbb{R}^{+}\rightarrow\mathbb{R}$). Given that every neighbourhood of $\{0\}$ contains an elements from every other equivalence class, the topology on the quotient space is a rather strange one. – Dan Rust Apr 27 '13 at 18:34

If you take the positive reals mod multiplication by $2$, then after taking logs this is isomorphic to the reals modulo addition by $\log 2$, so this is a copy of the circle. So your quotient is the union of the cirlce and one additional point.

Let's try to describe its topology:

Since the positive reals are open in the non-negative reals, and are closed under forming orbits with respect to $\sim$, we see that the circle is open subset of the quotient, and so the extra point is closed.

On other hand, if $x$ is any positive real, then $2^{-n} x$ converges to $0$. These points all have the same image modulo your equivalence, and so none of the points in the circle are closed (!) : they all contain the extra point in their closure.

So your quotient space is obtained by taking the circle, adding one closed point, and defining the open sets to be: the circle and its open subsets, together with the entire space.

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.