A note onR 1-topological spaces

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 [2]. 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.