Çetin Vural

SOME WEAKER FORMS OF THE CHAIN ( F ) CONDITION FOR METACOMPACTNESS

We define, in a slightly unusual way, the rank of a partially ordered set. Then we prove that if X is a topological space andW = {W(x) : x ∈ X} satisfies condition (F) and, for every x ∈ X ,W(x) is of the form ⋃ i∈n(x)Wi (x), whereW0(x) is Noetherian of finite rank, and every otherWi (x) 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 B of point-finite rank, then X is metacompact’, which was proved by Gruenhage and Nyikos. 2000 Mathematics subject classification: primary 54D20; secondary 03E02.

On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces

We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W0(x) ∪ W1(x), where W0(x) is Noetherian and W1(x) consists of neighbourhoods of x, then X is metacompact.