Alexander Subbase Theorem for Filters
Iwo Labuda · Bulletin of the Polish Academy of Sciences Mathematics · 2006
Presented by Czesªaw BESSAGA Summary.The theorem in the title is proven.Applications to product theorems are given.We adopt the terminology of [17℄, and so a topological space X = (X, O) is compact whenever one (and therefore all) of the following equivalent conditions is satised.• Every open cover of X contains a nite subcover;• Every lter on X has a cluster point;• Every ultralter on X is convergent.We note that, in contrast to [5℄ or [14℄, Hausdorness is not presupposed.The following [17, Ch. 5, Theorem 6℄ is the Alexander Subbase Theorem: AST.Let S be a subbase of O.If any cover of X by elements of S contains a nite subcover, then X is compact.We are interested in a more general form of AST in which the space X is replaced by a lter base or, equivalently, a lter.We need to introduce some terminology.Let P, H be families of subsets of X.We write P # H and say that P meshes with H if P ∩ H = ∅ for each P ∈ P and each H ∈ H.We say that H is a cover (resp.undercover) of a set A ⊂ X if A ⊂ H (resp.A ⊂ H = {H : H ∈ H}).A lter in O is a nonempty subfamily G ⊂ O which does not contain the empty set, is stable under nite intersections and such that if G ∈ G and G ⊂ H ∈ O, then H ∈ G.A lter which is a maximal element with respect 2000