On some applications of property (A) ((σ-A)) at a point

Abstract

If X is a hereditarily metacompact ω-scattered space and X has a σ-NSR pair-base at every point of X, then X has a σ-NSR pair-base. If X is a hereditarily meta-Lindelöf ω-scattered space and X has a σ-NSR pair-base at every point of X, then X has property (σ-A). If X is a hereditarily meta-Lindelöf GO-space such that every condensation set of X has property (σ-A), then X has property (σ-A). We point out that there is a gap in the proof of Lemma 37 in [18]. We give a detailed proof for the result. We finally show that if (X,τ,<) is a GO-space and X(n) has property (A) for some n∈N, then X has property (A), where X(0)=X, X(i+1)={x∈X(i):x is not an isolated point of X(i)} for each i<n. If X is a hereditarily meta-Lindelöf ω-scattered GO-space, then X has a σ-NSR pair-base and Xω is hereditarily a D-space.