Pre-Hausdorff Spaces

TL;DR

This paper introduces new separation conditions called pre-Hausdorff spaces, generalizing classical separation axioms, and explores their properties, characterizations, and implications for classical theorems in topology.

Contribution

It defines and characterizes pre-Hausdorff spaces, providing insights into their relation to Hausdorff spaces and classical topology theorems.

Findings

Pre-Hausdorff spaces generalize T1 and T2 separation axioms.

Characterizations of pre-Hausdorff spaces are established.

Some classical theorems are extended or limited when replacing Hausdorff with pre-Hausdorff.

Abstract

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.