When I read a math book, the book says
$\mathbb{R}/\mathbb{Z}$ is the set of coset $\mathbb{Z}$ in $\mathbb{R}$ with quotient topology induced by the usual topology on $\mathbb{R}$. The topology is also given by the metric $$ d(r+\mathbb{Z},s+\mathbb{Z}):=\min_{m \in \mathbb{Z}}|r-s+m| $$
so my question is why this metric can generate quotient topology in $\mathbb{R}/\mathbb{Z}$, this book does not explain anything, I know the $\mathbb{R}/\mathbb{Z}$ is homeomorphic to circle $S^1$( the map is $\pi(x+\Bbb Z) = e^{2\pi ix}$), so I want to use metric in $S^1$ to induce the metric in $\mathbb{R}/\mathbb{Z}$, but it does not work.
Another way of thinking about this: Let $x \in \mathbb R$ so there is a unique integer $\lfloor x \rfloor$ so that $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$. Let $\{x\}$ be a unique real so that $0\le \{x\} < 1$ and $x = \lfloor x \rfloor + \{x\}$.
Then the relationship that $y \sim x$ defined by $y \sim x \iff \{y\} = \{x\}$ is an equivalence. And $\frac {\mathbb R}{\mathbb Z}$ is the set of all equivalence classes.
Now $d(r+\mathbb{Z},s+\mathbb{Z}):=\min_{m \in \mathbb{Z}}|r-s+m|$ is the the same thing as $d(r+\mathbb{Z},s+\mathbb{Z}) = \begin{cases} |\{r\}-\{s\}| & |\{r\}-\{s\}| \le \frac 12 \\ \min(\{r\},\{s\}) + (1- \max(\{r\},\{s\})) &|\{r\}-\{s\}| >\frac 12 \end{cases}$
i.e. the shortest distance between $\{r\}$ and $\{s\}$ or the "next occurence".
I'm not really sure what you metric an $S^1$ was but I imagine it was similar, namely, the shortest distance along the surface of a circle between the two points.