TY - JOUR
T1 - Subcompactness and domain representability in GO-spaces on sets of real numbers
AU - Bennett, Harold
AU - Lutzer, David
N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.
PY - 2009/2/15
Y1 - 2009/2/15
N2 - In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if τ is any GO-topology on the real line R, then (R, τ) is subcompact, and so is any Gδ-subspace of (R, τ). We also show that if (X, τ) is a subcompact GO-space constructed on a subset X ⊆ R, then X is a Gδ-subset of any space (R, σ) where σ is any GO-topology on R with τ = σ |X. It follows that, for GO-spaces constructed on sets of real numbers, subcompactness is hereditary to Gδ-subsets. In addition, it follows that if (X, τ) is a subcompact GO-space constructed on any set of real numbers and if τS is the topology obtained from τ by isolating all points of a set S ⊆ X, then (X, τS) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known. We use our results on subcompactness to begin the study of other strong completeness properties in GO-spaces constructed on subsets of R. For example, examples show that there are subcompact GO-spaces constructed on subsets X ⊆ R where X is not a Gδ-subset of the usual real line. However, if (X, τ) is a dense-in-itself GO-space constructed on some X ⊆ R and if (X, τ) is subcompact (or more generally domain-representable), then (X, τ) contains a dense subspace Y that is a Gδ-subspace of the usual real line. It follows that (Y, τ |Y) is a dense subcompact subspace of (X, τ). Furthermore, for a dense-in-itself GO-space constructed on a set of real numbers, the existence of such a dense subspace Y of X is equivalent to pseudo-completeness of (X, τ) (in the sense of Oxtoby). These results eliminate many pathological sets of real numbers as potential counterexamples to the still-open question: "Is there a domain-representable GO-space constructed on a subset of R that is not subcompact"? Finally, we use our subcompactness results to show that any co-compact GO-space constructed on a subset of R must be subcompact.
AB - In this paper we explore a family of strong completeness properties in GO-spaces defined on sets of real numbers with the usual linear ordering. We show that if τ is any GO-topology on the real line R, then (R, τ) is subcompact, and so is any Gδ-subspace of (R, τ). We also show that if (X, τ) is a subcompact GO-space constructed on a subset X ⊆ R, then X is a Gδ-subset of any space (R, σ) where σ is any GO-topology on R with τ = σ |X. It follows that, for GO-spaces constructed on sets of real numbers, subcompactness is hereditary to Gδ-subsets. In addition, it follows that if (X, τ) is a subcompact GO-space constructed on any set of real numbers and if τS is the topology obtained from τ by isolating all points of a set S ⊆ X, then (X, τS) is also subcompact. Whether these two assertions hold for arbitrary subcompact spaces is not known. We use our results on subcompactness to begin the study of other strong completeness properties in GO-spaces constructed on subsets of R. For example, examples show that there are subcompact GO-spaces constructed on subsets X ⊆ R where X is not a Gδ-subset of the usual real line. However, if (X, τ) is a dense-in-itself GO-space constructed on some X ⊆ R and if (X, τ) is subcompact (or more generally domain-representable), then (X, τ) contains a dense subspace Y that is a Gδ-subspace of the usual real line. It follows that (Y, τ |Y) is a dense subcompact subspace of (X, τ). Furthermore, for a dense-in-itself GO-space constructed on a set of real numbers, the existence of such a dense subspace Y of X is equivalent to pseudo-completeness of (X, τ) (in the sense of Oxtoby). These results eliminate many pathological sets of real numbers as potential counterexamples to the still-open question: "Is there a domain-representable GO-space constructed on a subset of R that is not subcompact"? Finally, we use our subcompactness results to show that any co-compact GO-space constructed on a subset of R must be subcompact.
KW - Amsterdam properties
KW - Co-compact
KW - Domain
KW - Domain-representable space
KW - GO-space
KW - Generalized ordered space
KW - Pseudo-complete space
KW - Strong Choquet game
KW - Subcompact space
UR - http://www.scopus.com/inward/record.url?scp=60249094930&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2008.11.012
DO - 10.1016/j.topol.2008.11.012
M3 - Article
AN - SCOPUS:60249094930
VL - 156
SP - 939
EP - 950
JO - Topology and its Applications
JF - Topology and its Applications
SN - 0166-8641
IS - 5
ER -