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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?

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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. –  Daniel Rust Apr 27 '13 at 18:34
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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.

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