A glance at spaces with closure-preserving local bases

Abstract

Call a space X ( weakly) Japanese at a point x ∈ X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is ( weakly) Japanese if it is (weakly) Japanese at every x ∈ X . We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have t ( A ¯ ) ⩽ | A | for any set A ⊂ X .