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.

$\mathbb{T}=\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$

Forward jump:	$\sigma(t)=$	derivation
Forward graininess:	$\mu(t)=$	derivation
Backward jump:	$\rho(t)=$	derivation
Backward graininess:	$\nu(t)=$	derivation
$\Delta$-derivative	$f^{\Delta}(t)=$	derivation
$\nabla$-derivative	$f^{\nabla}(t)=$	derivation
$\Delta$-integral	$\displaystyle\int_s^t f(\tau) \Delta \tau=$	derivation
$\nabla$-integral	$\displaystyle\int_s^t f(\tau) \nabla \tau=$	derivation
$h_k(t,s)$	$h_k(t,s)=$	derivation
$\hat{h}_k(t,s)$	$\hat{h}_k(t,s)=$	derivation
$g_k(t,s)$	$g_k(t,s)=$	derivation
$\hat{g}_k(t,s)$	$\hat{g}_k(t,s)=$	derivation
$e_p(t,s)$	$e_p(t,s)=$	derivation
$\hat{e}_p(t,s)$	$\hat{e}_p(t,s)=$	derivation
Gaussian bell	$\mathbf{E}(t)=$	derivation
$\mathrm{sin}_p(t,s)=$	$\sin_p(t,s)=$	derivation
$\mathrm{\sin}_1(t,s)$	$\sin_1(t,s)=$	derivation
$\widehat{\sin}_p(t,s)$	$\widehat{\sin}_p(t,s)=$	derivation
$\mathrm{\cos}_p(t,s)$	$\cos_p(t,s)=$	derivation
$\mathrm{\cos}_1(t,s)$	$\cos_1(t,s)=$	derivation
$\widehat{\cos}_p(t,s)$	$\widehat{\cos}_p(t,s)=$	derivation
$\sinh_p(t,s)$	$\sinh_p(t,s)=$	derivation
$\widehat{\sinh}_p(t,s)$	$\widehat{\sinh}_p(t,s)=$	derivation
$\cosh_p(t,s)$	$\cosh_p(t,s)=$	derivation
$\widehat{\cosh}_p(t,s)$	$\widehat{\cosh}_p(t,s)=$	derivation
Gamma function	$\Gamma_{\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}}(x,s)=$	derivation
Euler-Cauchy logarithm	$L(t,s)=$	derivation
Bohner logarithm	$L_p(t,s)=$	derivation
Jackson logarithm	$\log_{\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}} g(t)=$	derivation
Mozyrska-Torres logarithm	$L_{\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}}(t)=$	derivation
Laplace transform	$\mathscr{L}_{\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}}\{f\}(z;s)=$	derivation
Hilger circle		derivation

Examples of time scales
$\Huge\mathbb{R}$ Real numbers	$\Huge\mathbb{Z}$ Integers	$\Huge{h\mathbb{Z}}$ Multiples of integers	$\Huge\mathbb{Z}^2$ Square integers	$\Huge\mathbb{H}$ Harmonic numbers	$\Huge\mathbb{T}_{\mathrm{iso}}$ Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$ nth root numbers	$\Huge\mathbb{P}_{a,b}$ Evenly spaced intervals	$\huge\overline{q^{\mathbb{Z}}}$ Quantum, $q>1$	$\huge\overline{q^{\mathbb{Z}}}$ Quantum, $q<1$	$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$ Closure of unit fractions	$\Huge\mathcal{C}$ Cantor set