  In this paper, we introduce the notions of Star-$\sigma\mathcal{K}$ and
absolutely Star-$\sigma\mathcal{K}$ spaces which allow us to unify results
among several properties in the theory of star selection principles on small
spaces. In particular, results on star selective versions of the Menger,
Hurewicz and Rothberger properties and selective versions of property $(a)$
regarding the size of the space. Connections to other well-known star
properties are mentioned. Furthermore, the absolute and selective version of
the neighbourhood star selection principle are introduced. As an application,
it is obtained that the extent of a separable absolutely strongly star-Menger
(absolutely strongly star-Hurewicz) space is at most the dominating number
$\mathfrak{d}$ (the bounding number $\mathfrak{b}$).
