Rudin's Dowker space, strong base-normality and base-strong-zero-dimensionality

It is well known that every pair of disjoint closed subsets F0,F1 of a normal T1-space X admits a star-finite open cover U of X such that, for every U∈U, either or holds. We define a T1-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 F0,F1 of X admits a star-finite cover B′ of X by members of B such that, for every B∈B′, either or holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudin's Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied.