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 W = {W(x) : x ∈ X } satisfies condition (F) and, for every x ∈ X , W(x) is of the form i∈n(x) W i (x), where W 0 (x) is Noetherian of finite rank, and every other W i (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.

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We prove that every retract of a domain representable space is domain representable. Consequently, we obtained that if C p (X) is a countable union of its closed subcompact subspaces then X is discrete. This solves Question 7 in [5].

Abstract: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 } 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 , W( ) is of the form W 0 ( ) ∪ W 1 ( ), where W 0 ( ) is Noetherian and W 1 ( ) consists of neighbourhoods of , then X is metacompact. MSC:54D20, 03E02

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