In this paper, we introduce and investigate three classes of subsets 
called $\xi$-closed sets, $\xi^{\ast}$-
closed sets and ${\xi}^{\ast \ast}$-closed sets in topological spaces.  
As applications
we introduce two separation axioms $T_{\xi}$ and $T_{\xi^{\ast \ast}}$ 
of topological spaces and we construct  a group of $\rho$c-homeomorphisms 
which contains the group of all homeomorphisms as a subgroup, 
where $\rho \in \{\xi, \xi ^{\ast}, \xi ^{\ast \ast}\}$.  
A discussion of $\rho$-closed sets in the digital plane 
concludes the paper, where $\rho \in \{\xi, \xi ^{\ast \ast}\}$.  
The digital plane is a $T_{\xi}$-space; it is not $T_{\xi ^{\ast \ast}}$.