Rudinʹs Dowker space, strong base-normality and base-strong-zero-dimensionality

It is well known that every pair of disjoint closed subsets F 0 , F 1 of a normal T 1 -space X admits a star-finite open cover U of X such that, for every U ∈ U , either U ¯ ∩ F 0 = ∅ or U ¯ ∩ F 1 = ∅ holds. We define a T 1 -space X to be strongly base-normal if there is a base B for X with | B | = w ( X ) satisfying that every pair of disjoint closed subsets F 0 , F 1 of X admits a star-finite cover B ′ of X by members of B such that, for every B ∈ B ′ , either B ¯ ∩ F 0 = ∅ or B ¯ ∩ F 1 = ∅ holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudinʹs Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied.