Spaces with σ-point finite bases

AbstractTheorem. Let X be a T1 space. The following are equivalent:(1)X has a σ-disjoint base.(FX2)X is quasi-developable and has a base that is the union of a sequence of rank 1 collections.(3)X has a quasi-development (Gn) with the property that for each x, {st2(x,Gn): x ∈st2(x,Gn), n a positive integer} is a base for N (x).Theorem. Let X be a T1 space. The following are equivalent:(1)X has a σ-point finite base.(2)X has a quasi-development (Gn) with each Gn well ranked.(3)X has a quasi-development (Gn) with each Gn Noetherian of sub-infinite rank.(4)X has a quasi-development (Gn) with each Gn Noetherian of point finite rank