This paper introduces three separation conditions for topological spaces, called T_{ 0,1}, T_{ 0,2} ("pre-Hausdorff"), and T_{ 1,2}. These conditions generalize the classical T_(1) and T_(2) separation axioms, and they have advantages over them topologically which we discuss. We establish several different characterizations of pre-Hausdorff spaces, and a characterization of Hausdorff spaces in terms of pre-Hausdorff. We also discuss some classical Theorems of general topology which can or cannot be generalized by replacing the Hausdorff condition by pre-Hausdorff.
Categories: math.GN