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

Önal, Süleyman
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 a X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of omega-rank, then X is metaLindelof as well as a countable product of them. We also prove that if W satisfies omega-rank (F) and, for every x a X, W(x) is of the form W (0)(x) a(a) W (1)(x), where W (0)(x) is Noetherian and W (1)(x) consists of neighbourhoods of x, then X is metacompact.