coverage mapping at covering spaces

Question

the professor at university asked us: is a coverage mapping like P from X to Y a closed mapping or not. Also; is p an open mapping? i could prove that P is an open mapping but for proving that P is not closed mapping; i should use the fact that if X equals to real line and Y equals to $S^1$ and the P domain order equal $n+1/n$ for $n>3$ but i cant prove it .would you help me please?

Answer 1

You didn't tell us which map you are talking about, but I suspect that it is$$\begin{array}{rccc}P\colon&\mathbb R&\longrightarrow&S^1\\&x&\mapsto&(\cos2\pi x,\sin2\pi x).\end{array}$$If that's so and if $S=\left\{n+\frac1n\,\middle|\,n\in\mathbb N\setminus\{1,2\}\right\}$, then $S$ is a closed subset of $\mathbb R$, but $P(S)$ is not a closed subset of $S^1$, since$$\lim_{n\to\infty}P\left(n+\frac1n\right)=0,$$but $0\notin P(S)$.