A metric space of a.h. stone and an example concerning σ-minimal bases

Harold R. Bennett; David J. Lutzer

doi:10.1090/s0002-9939-98-04785-6

A metric space of a.h. stone and an example concerning σ-minimal bases

Harold R. Bennett, David J. Lutzer

Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

In this paper we use a metric space V due to A. H. Stone and one of its completions X to construct a linearly ordered topological space E = E(Y,X) that is Čecil complete, has a tr-closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a σ-minimal base for its relative topology. However, E is not metrizable and is not quasi-developable. The construction of E(Y, X) is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone's metric space and our construction.

Original language	English
Pages (from-to)	2191-2196
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	126
Issue number	7
DOIs	https://doi.org/10.1090/s0002-9939-98-04785-6
State	Published - 1998

Keywords

Cech complete
Cr-minimal base
Generalized ordered space
Linearly ordered space
Metrization theory
Paracompact
Perfect space

Access to Document

10.1090/s0002-9939-98-04785-6

Fingerprint Dive into the research topics of 'A metric space of a.h. stone and an example concerning σ-minimal bases'. Together they form a unique fingerprint.

Cite this

@article{b26affd14a924bd7a02a281765fb2d5d,

title = "A metric space of a.h. stone and an example concerning σ-minimal bases",

abstract = "In this paper we use a metric space V due to A. H. Stone and one of its completions X to construct a linearly ordered topological space E = E(Y,X) that is {\v C}ecil complete, has a tr-closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a σ-minimal base for its relative topology. However, E is not metrizable and is not quasi-developable. The construction of E(Y, X) is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone's metric space and our construction.",

keywords = "Cech complete, Cr-minimal base, Generalized ordered space, Linearly ordered space, Metrization theory, Paracompact, Perfect space",

author = "Bennett, {Harold R.} and Lutzer, {David J.}",

year = "1998",

doi = "10.1090/s0002-9939-98-04785-6",

language = "English",

volume = "126",

pages = "2191--2196",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

number = "7",

}

TY - JOUR

T1 - A metric space of a.h. stone and an example concerning σ-minimal bases

AU - Bennett, Harold R.

AU - Lutzer, David J.

PY - 1998

Y1 - 1998

N2 - In this paper we use a metric space V due to A. H. Stone and one of its completions X to construct a linearly ordered topological space E = E(Y,X) that is Čecil complete, has a tr-closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a σ-minimal base for its relative topology. However, E is not metrizable and is not quasi-developable. The construction of E(Y, X) is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone's metric space and our construction.

AB - In this paper we use a metric space V due to A. H. Stone and one of its completions X to construct a linearly ordered topological space E = E(Y,X) that is Čecil complete, has a tr-closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a σ-minimal base for its relative topology. However, E is not metrizable and is not quasi-developable. The construction of E(Y, X) is a point-splitting process that is familiar in ordered spaces, and an orderability theorem of Herrlich is the link between Stone's metric space and our construction.

KW - Cech complete

KW - Cr-minimal base

KW - Generalized ordered space

KW - Linearly ordered space

KW - Metrization theory

KW - Paracompact

KW - Perfect space

UR - http://www.scopus.com/inward/record.url?scp=22044446275&partnerID=8YFLogxK

U2 - 10.1090/s0002-9939-98-04785-6

DO - 10.1090/s0002-9939-98-04785-6

M3 - Article

AN - SCOPUS:22044446275

VL - 126

SP - 2191

EP - 2196

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 7

ER -

A metric space of a.h. stone and an example concerning σ-minimal bases

Abstract

Keywords

Access to Document

Other files and links

Cite this