The questions are the following:
Consider the five topologies on the real line $\mathbb R$:
$\mathcal T_1$: the standard topology
$\mathcal T_2$: the $K$-topology
$\mathcal T_3$: the finite complement topology
$\mathcal T_4$: the upper limit topology
$\mathcal T_5$: the topology generated by the basis $\{(-\infty,a)\mid a \in \mathbb R\}$
Determine the closure of the set $K=\{\frac{1}{n}\mid n\in\mathbb N\}$ under each of these topologies.
My answer is the following:
$\mathcal T_1$: $\mathrm{cl}(K)=\{0\} \cup K$.
$\mathcal T_4$: $\{0\} \cup K$.
$\mathcal T_5$: $[0, \infty)$
Thank you.