Functional compactifications of $T_0$-spaces and bitopological structures

We study the compactifcation of $T_0$-spaces generated by families of special continuous mappings into a given standard space $E$. In this context we have introduced the notions of $E$-thin and $E$-rough $g$-compactifications. The maximal $E$-thin and $E$-rough $g$-compactifications are constructed.