Let $f: \mathbb{R} \to S^1$ defined $$f(t)=(\cos(2\pi t), \sin (2\pi t))$$ I have to prove that the quotient topology $\mathcal{U}_f$ induced by $f$ on $S^1$ is equal to the topology $\mathcal{U}$ on $S^1$ as a subespace of $\mathbb{R}^2$.
This is my idea let $V \in \mathcal{U}$ then $V= S^1 \cap U$ where $U$ is open on $\mathbb{R}^2$, and I have to prove that $f^{-1} (V) $ is open in $\mathbb{R}$.
The book gives the suggestion to prove that $(S^1, \mathcal{U}_f)$ is homeomorphic to $(S^1, \mathcal{U})$
This follows from the universal property of quotient maps. All you need to show is that $f$ is a quotient map. And it is, because it is open as you can see here.