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摘要
Properties (A) and (B) were introduced by Chase and Gruenhage in 2016. Let X be a topological space and let P(X) be the collection of all pairs (x, U) where x is an element of X and U is an open neighborhood of x in X. A space X has property (A) (resp., property (B)) if to each (x, U) is an element of P(X), one can assign an open set V(x, U) such that x is an element of V(x, U) subset of U and such that for any collection {(x(alpha), U-alpha : alpha is an element of A} subset of P(X), either boolean AND V-alpha is an element of A (x(alpha), U-alpha) = empty set, or there exists a finite set A ' subset of A such that for each alpha is an element of A there exists beta is an element of A' with V(x(alpha), U-alpha) subset of U-beta (resp., {x(alpha): alpha is an element of A } subset of U-beta is an element of A' U-beta). We discuss properties of property (A) ((B)) and get the following conclusions: If a regular topological space X has caliber omega(l) 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, tau, <) is a GO-space such that X \ I-tau has property (A) ((B)), then X has property (A) ((B)), where I-tau = {x is an element of X : {x} is open in X}. If (X,tau, <) is a GO-space whose underlying LOTS (X, lambda, <) has a sigma-closed-discrete dense subset, then X is monotonically metacompact if and only if X \ I-tau is monotonically countably metacompact, where I-tau = {x is an element of X : {x} is open in X}.
- 出版日期2018-8-15
- 单位北京工业大学
