We give a survey of the known results concerning the sets $c_0(\Lambda)$, $c(\Lambda)$ and $c_\infty(\Lambda)$ including their basic topological properties, their first and second dual spaces, and the characterizations of matrix transformations from them into the spaces $\ell_\infty$, $c$ and $c_0$. Furthermore, we establish some new results such as the representations of the general bounded linear operators from $c(\Lambda)$ into the spaces $\ell_\infty$, $c$ and $c_0$, and estimates for their Hausdorff measures of noncompactness. Finally, we apply our results to characterize some classes of compact operators on $c_0(Lambda)$, $c(Lambda)$ and $c\infty(Lambda)$. We also generalize a classical result by Cohen and Dunford which states that a regular matrix operator cannot be compact.