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

Süleyman Önal 1  and Çetin Vural 2
  • 1 Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey
  • 2 Fen Fakultesi, Matematik Bolumu, Gazi Universitesi, 06500, Teknikokullar, Ankara, Turkey

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): 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 W 0(x) ∪ W 1(x), where W 0(x) is Noetherian and W 1(x) consists of neighbourhoods of x, then X is metacompact.

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