Published online by Cambridge University Press: 01 April 2008
We define, in a slightly unusual way, the rank of a partially ordered set. Then we prove that if X is a topological space and satisfies condition (F) and, for every x∈X, is of the form , where is Noetherian of finite rank, and every other is a chain (with respect to inclusion) of neighbourhoods of x, then X is metacompact. We also obtain a cardinal extension of the above. In addition, we give a new proof of the theorem ‘if the space X has a base of point-finite rank, then X is metacompact’, which was proved by Gruenhage and Nyikos.