Let $X=S^1\times I$ where $S^1\subset \mathbb{C}$ and $I=[0,1]\subset \mathbb{R}$ also define the equivalence relation ~ by $(x,t)$~$(y,s)$ if and only if $xt=ys$.
Prove that $X$/~ is homeomorphic to the unit disc $D^2=\{x\in \mathbb{R^2}: |x|\le1\}=\{x\in \mathbb{C}: |x|\le1\}$ with the induced topology.
Any hints would be appreciated.
Consider the function $f\colon S^1 \times I \to D^2$ defined by $f(x, t) = xt$. Show $f(x, t) = f(y, s)$ iff the pairs are equivalent, so $f$ induces a function on the quotient space
$$\bar{f}\colon (S^1 \times I)/\sim \to D^2 $$
Now show that this function is a homeomorphism.