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 $(X,\tau ,<)$ is a GO-space and $X^{\left(n\right)}$ has property (A) for some $n\in N$, then X has property (A), where $X^{\left(0\right)}=X$, $X^{(i+1)}=\{x\in X^{\left(i\right)}:x$ is not an isolated point of $X^{\left(i\right)}\}$ for each $i<n$. If X is a hereditarily meta-Lindelöf ω-scattered GO-space, then X has a σ-NSR pair-base and $X^{\omega }$ is hereditarily a D-space.