$S^1$ is the subspace of $\mathbb{R}^2$ consisting of all $y\in\mathbb{R}^2$ such that $d(y,0)=1$, where $d$ is the Eucledian metric. Let $f:\mathbb{R}\to S^{1}$ be given by $f(x)=(\cos(2\pi x),\sin(2\pi x)))$. Show that $f$ is the quotient mapping but not a closed mapping. Is $f$ an open mapping?
I have no idea on how to solve this question. A closed set in $S_1$ is by the definition of subspace topological space the intersection of a closed subset of $\mathbb{R}^2$. But I cannot see how I can relate all these concepts to the quotient topology.
Question:
Can someone help me solve this question?
Thanks in Advance!
It is not a closed map, because$$S=\left\{n+\frac1n\,\middle|\,n\in\mathbb N\setminus\{1\}\right\}$$is closed, but $f(S)$ isn't ($(1,0)\in\overline{f(S)}\setminus f(S)$).