What are the topological properties of the quotient space $X$ obtained from $\mathbb{C}$ by identifying points of the same modulus? I.e., the space $X=\mathbb{C}/E$ where $E$ is the equivalence relation $x\sim y$ iff $|x|=|y|$. Is this space homeomorphic to any commonly known topological space? What if we remove a countable dense subset from $\mathbb{C}$, obtaining a space we call $\mathbb{C}_0$ and ask the same question for $X_0=\mathbb{C}_0/E_0$, where $E_0$ is $E$ restricted to $\mathbb{C}_0$?

[Added April 14, 2015] This question is a particular instance of a more general problem which I am trying to understand. Suppose $Y$ is a Polish space (i.e., a complete separable metric space) and $E$ is an equivalence relation on $Y$ such that every equivalence class is homeomorphic to the unit circle. Do we know that the number of equivalence classes is either finite, countably infinite, or $\ \mathfrak{c}$? My illustrative example turns out to be trivial and not very helpful. I suspect the answer in general may depend on CH. In ZFC we can get a YES if we assume more about the equivalence relation, e.g., it is a closed (or even coanalytic) subset of the product space $Y\times Y$. I suspect the stated assumptions on $E$ do not imply this condition.