θ-regular spaces - Dragan S. Janković (International Journal of Mathematics and Mathematical Sciences)
In this paper we define a topological space X to be θ-regular if every filterbase in X with a nonempty θ-adherence has a nonempty adherence. It is shown that the class of θ-regular topological spaces includes rim-compact topological spaces and that θ-regular H(i) (Hausdorff) topological spaces are compact (regular). The concept of θ-regularity is used to extend a closed graph theorem of Rose [1]. It is established that an r-subcontinuous closed graph function into a θ-regular topological space is continuous. Another sufficient condition for continuity of functions due to Rose [1] is also extended by introducing the concept of almost weak continuity which is weaker than both weak continuity of Levine and almost continuity of Husain. It is shown that an almost weakly continuous closed graph function into a strongly locally compact topological space is continuous.