\(\theta\)-regular spaces.

*(English)*Zbl 0577.54012Summary: A topological space X is called \(\theta\)-regular if every filterbase in X with a nonempty \(\theta\)-adherence has a nonempty adherence. It is shown that the class of \(\theta\)-regular spaces includes rim-compact spaces and that \(\theta\)-regular H(i) (Hausdorff) spaces are compact (regular). The concept of \(\theta\)-regularity is used to extend a closed graph theorem of D. A. Rose [Can. Math. Bull. 21, 477-481 (1978; Zbl 0394.54004)]. It is established that an r-subcontinuous closed graph function into a \(\theta\)-regular space is continuous. Another sufficient condition for continuity of functions due to Rose (loc. cit.) 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 space is continuous.

##### MSC:

54C10 | Special maps on topological spaces (open, closed, perfect, etc.) |

54C08 | Weak and generalized continuity |

54D45 | Local compactness, \(\sigma\)-compactness |