$E=\left\{ \frac{n}{n+1}:n\in\mathbb{N}\right\} $and $F=\left\{ \frac{1}{1-x}:0\leq x<1\right\} $ is open or closed?

Question

QuestionWhich of The following is open or closed $E=\left\{ \frac{n}{n+1}:n\in\mathbb{N}\right\} $and F=$\left\{ \frac{1}{1-x}:0\leq x<1\right\} $

MY Approach

Let $f$ : $E\longrightarrow\mathbb{R}$defined by $f\left(\frac{n}{n+1}\right)=n$

I do not know ,It is continuous or not But if yes ,then E is closed

Help me , how to define a continuous function to know that set is open or closed?

Answer 1

$E$ is not closed since $1$ is an accumulation point of $E$ which does not belong to $E$.

For the set $F$, actually $F=[1,\infty)$, so $F$ is closed.