# Closure of unit fractions

The set $\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}=\left\{ 0,1,\dfrac{1}{2},\dfrac{1}{3},\ldots \right\}$, where the $\overline{\mathrm{overline}}$ denotes topological closure of this set in the usual topology on $\mathbb{R}$ is a time scale.
 Generic element $t\in \mathbb{T}$: Either $t=0$ or for some $n \in \mathbb{Z}^+, t = \dfrac{1}{n}$ Jump operator: $\sigma(t) = \left\{ \begin{array}{ll} \dfrac{1-t}{t} &; t>1 \\ 0 &; t=0 \end{array} \right.$ Graininess operator: $\mu(t) = \left\{ \begin{array}{ll} \dfrac{1-t-t^2}{t} &; t>0 \\ 0 &; t=0 \end{array} \right.$ $\Delta$-derivative: $f^{\Delta}(t) = \left\{ \begin{array}{ll} \dfrac{t}{1-t} \left[ f \left(\dfrac{1-t}{t} \right)-f(t) \right] &; t>0 \\ \displaystyle\lim_{h \rightarrow 0} \dfrac{f(h)-f(0)}{h} &; t=0 \end{array} \right.$ $\Delta$-integral: Exponential function: