R 1 -Topological Spaces 1

Abstract

In his paper ″Indexed systems of neighborhoods for general topological spaces″ (Amer. Math. Monthly 68, (1961), 886–893), A.S. Davis defined a hierarchy of what he called regularity axioms. The R 1 -axiom is independent of both T 0 and T 1 , but is strictly weaker than T 2 . In this note, we propose to study the properties of the spaces satisfying the R 1 -axiom. In particular, we will show that in many well-known results, the hypothesis can be weakened from T 2 to R 1 , which is part of our motivation in studying R 1 -spaces.