One of Munkres' exercises tasks us to consider the following quotient space $\mathbb{R}^2/\sim$ in $\mathbb{R}^2$ given by the following equivalence relationship: $$(x,y)\sim(a,b)\iff x^2 + y^2 = a^2 + b^2$$
If one's to consider $\mathbb{R}^2/\sim$ graphically, it's composed by all the circles whose center is at $(0,0)$. The exercise asks: "this quotient space is homeomorphic to a well-known space, which is it?". I incline to think that we can map a circle to its radius so, for me, $$\mathbb{R}^2/_\sim \; \simeq \; [0,\infty)$$ For this, I consider $f\colon [0,\infty) \to \mathbb{R}^2/\sim$ with $r\mapsto [(0,r)]$ (i.e. the circle with radius $r$). I must prove that $f$ is an homeomorphism, but I'm unable to grasp how the open sets in the quotient topology look like. Would anyone give me some insight on how these open sets look like and how would I work with them?
For a set $U\in X/{\sim}$ to be open, you require that the union of the equivalence classes $$\bigcup_{[x]\in U}[x]$$ be open in $X$.
What they look like. So in this case, you require that the union of the equivalence classes be an open subset of $\mathbb{R}^2$ which, as AJ Stas mentions in the comments, are open annuli.
How to work with them. Prove that the annuli $$\{[(x,y)]\in\mathbb{R}^2\;\colon\;a<x^2+y^2<b\},\quad a<b$$ form a basis of the topology on $\mathbb{R}^2/{\sim}$; and that their pre-images under (your) $f$ are open intervals $(a,b)\in[0,\infty)$, and you have shown $f$ is continuous.