A study on monotonically metacompact and property (A) ((B))

We discuss properties of property (A) ((B)) and get the following conclusions: If a regular topological space X has caliber ω1 and satisfies property (A), then X is a second countable metrizable space. If X is a GO-space, then X has property (A) ((B)) if and only if the closed linearly ordered extension X⁎ of X has property (A) ((B)). If X is a scattered GO-space which satisfies property (B), then X is monotonically metacompact. If L is a monotonically metacompact GO-space and Y is a convex subspace of L, then Y is monotonically metacompact. If (X,τ,<) is a GO-space such that X∖Iτ has property (A) ((B)), then X has property (A) ((B)), where Iτ={x∈X:{x} is open in X}. If (X,τ,<) is a GO-space whose underlying LOTS (X,λ,<) has a σ-closed-discrete dense subset, then X is monotonically metacompact if and only if X∖Iτ is monotonically countably metacompact, where Iτ={x∈X:{x} is open in X}.