To find the closure of a set with respect to different topologies on R.

Question

Find the closure of K=$\{\frac{1}{n}|n\in N\}$ with usual standard topology on $\mathbb R$ and with finite complement topology on $\mathbb R$.

For standard topology on R , clearly I got Closure of K =$\{0, 1 , 1/2 ,1/3,...\}$.

But for finite complement topology on $\mathbb R$, I don't know how to proceed. $T =\{ U \subseteq R| R\setminus U\text{ is finite or all of }\mathbb R\}$ is called finite complement topology on $\mathbb R$.

Answer 1

Note that in the cofinite topology, or the finite complement topology, closed sets are either finite sets or the whole space by definition. Thus the only closed set containing $K$, which has infinitely many elements, must be $\Bbb R$. Hence the closure must be $\Bbb R$.