Let $\sim$ be and equivalence relation on the unit line $X=[0,1]$ defined by $x\sim y$ if either $x=y$ or $\textbf{both}$ $x$ and $y$ $\in$ {${0,\frac{1}{2},1}$}.
Construct a homeomorphism $f:\frac{X}{\sim}\to Y$ between the identification space $\frac{X}{\sim}$ and the topological space $Y=C_+\cup C_- \subset \mathbb{R}^2$ with $C_+$ the circle with centre $(1,0)$ and radius 1, and $C_-$ the circle with centre $(-1,0)$ and radius 1. Y is considered with the subspace topology.
Edit: This is a past paper question to an upcoming Topology exam paper that I have to sit, unfortunately no solutions were provided for the question. I get that I'm identifying the points $(1,\frac{1}{2})$ $(\frac{1}{2}, 0)$ etc. with each other, but not sure where to go from there. Since it's mapped to $C_+\cup C_-$ I have a feeling that it could be some sort of $t \mapsto (cos2\pi t, sin2\pi t)$ solution but very confused!
HINT: Map the point $\left\{0,\frac12,1\right\}$ to the origin, i.e., to the point where the two circles are tangent. The intervals $\left(0,\frac12\right)$ and $\left(\frac12,1\right)$ will then map to ... what? (Picture starting with $[0,1]$ and bending the ends around so that they can both be glued to the point $\frac12$.)