Weakly Perfect Generalized Ordered Spaces

A space X is weakly perfect if each closed subset of X contains a dense subset that is a Gδ-subset of X. This property was introduced by KocÌ\u8cinac and later studied by Heath. We provide three mechanisms for constructing ZFC examples of spaces that are weakly perfect but not perfect. Some of our examples are compact linearly ordered spaces, while others are types of Michael lines. Our constructions begin with special subsets of the usual unit interval, e.g., perfectly meager subsets. We conclude by giving a new and strictly internal topological characterization of perfectly meager subsets of [0, 1], namely that a topological space X is homeomorphic to a perfectly meager subset of [0, 1] if and only if X is a zero-dimensional separable metrizable space with the property that every subset A â\u8a\u82 X contains a countable set B that is dense in A and is a Gδ-subset of X