TY - JOUR
T1 - Weakly perfect generalized ordered spaces
AU - Bennett, Harold R.
AU - Hosobuchi, Masami
AU - Lutzer, David J.
N1 - Copyright:
Copyright 2004 Elsevier Science B.V., Amsterdam. All rights reserved.
PY - 2000
Y1 - 2000
N2 - 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 Kočinac 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 ⊂ X contains a countable set B that is dense in A and is a Gδ-subset of X.
AB - 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 Kočinac 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 ⊂ X contains a countable set B that is dense in A and is a Gδ-subset of X.
KW - Baire category
KW - Generalized ordered space
KW - Lexicographic product
KW - Linearly ordered space
KW - Perfect space
KW - Perfectly meager subset
KW - Weakly perfect
UR - http://www.scopus.com/inward/record.url?scp=0040784666&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:0040784666
VL - 26
SP - 609
EP - 627
JO - Houston Journal of Mathematics
JF - Houston Journal of Mathematics
SN - 0362-1588
IS - 4
ER -