Chapter 2.17, Problem 16.1E

To determine

To find:

The closure of the set $K=\left\{1/n|n\in {\mathbb{Z}}_{+}\right\}$ under each of these topologies.

$T_1=$ the standard topology,

$T_2=$ the topology of ${ℝ}_K$,

$T_3=$ the finite complement topology,

$T_4=$ the upper limit topology, having the sets $\left(a,b\right]$ as basis,

$T_5=$ the topology having all sets $\left(-\infty ,a\right)=\left\{x|x<a\right\}$ as basis.