This was proposed by my lecturer as an excercise with no grade at all. However, I found it tricky to understand and solve.
Given the partition $\mathcal{D}=\{c_r|r\in[0,\infty)\}$ in $\mathbb{R}^2$, where $c_r$ is the circumference with radius $r$ and center at the origin, prove $(\mathcal{D},\tau_\mathcal{D})$ is homeomorphic to $[0,\infty)$ as a subspace of $(\mathbb{R},\tau_\mathbb{R})$ ($\tau_\mathbb{R}$ the usual topology).
I have made my research, and found that a circumference cannot be homeomorphic to any open interval in $\mathbb{R}$ so, I do not know how to start from here and find a function $f$ to give the homeomorphism.