On increasing separated sequences

In this paper, we investigate some generalizations of normality and collectionwise normality. In particular, we answer a question of Balogh and Burke by showing that if there is a collectionwise normal space that is not countably metacompact, then there is a normal ↗-collectionwise normal space that is not collectionwise normal. The proof is an application of a machine constructed by Rudin in [Fund. Math. 73 (1971) 179–186].