The purpose of the present note is to give a number of characterizations of theR1-axiom and to show that theR1-axiom is equivalent to the weakly Hausdorff axiom introduced byB. Banaschewski andJ. M. Maranda . In anR1-space it is shown that the locally compactness property is also open hereditary and that the closure of an almost compact set is the union of the closures of its points. A necessary and sufficient condition is obtained under which a locally compact set dense in anR1-space is open. Finally a variant of a well-known theorem regarding two continuous functions of a topological space into aT2-space is formulated forR1-spaces.
In a topological spaceX, a T2-distinct pointx means that for anyy∈X x≠y, there exist disjoint open neighbourhoods ofx andy. Similarly, T0-distinct points and T1distinct points are defined. In a Ti-distinct point-setA, we assume that eachx∈A is a Ti-distinct point (i=0, 1, 2). In the present paper some implications of these notions which ‘localize’ the Ti-separation axioms (i=0, 1, 2) requirement, are studied. Suitable variants of regularity and normality in terms of T2-distinct points are shown hold in a paracompact space (without the assumption of any separation axioms). Later T0-distinct points are used to give two characterizations of the RD-axiom.1 In the end, some simple results are presented including a condition under which an almost compact set is closed and a result regarding two continuous functions from a topological space into a Hausdorff space is sharpened. A result which relates a limit pointv to an ω-limit point is stated.