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$$\theta$$-regular spaces. (English) Zbl 0577.54012
Summary: 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
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