A non-metrizable compact

linearly ordered topological space,

every subspace of which

has a -minimal base

Author:
Wei-Xue Shi

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2783-2791

MSC (1991):
Primary 54F05, 54G20, 54E35

DOI:
https://doi.org/10.1090/S0002-9939-99-04853-4

Published electronically:
April 15, 1999

MathSciNet review:
1600133

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A collection of subsets of a space is minimal if each element of contains a point which is not contained in any other element of . A base of a topological space is -minimal if it can be written as a union of countably many minimal collections. We will construct a compact linearly ordered space satisfying that is not metrizable and every subspace of has a -minimal base for its relative topology. This answers a problem of Bennett and Lutzer in the negative.

**1.**C. E. Aull,*Quasi-development and -base,*J. London Math. Soc. (2), 9(1974), 197-204. MR**52:9171****2.**H. R. Bennett and E. S. Berney,*Space with -minimal base,*Topology Proceedings 2(1977), 1-10. MR**80k:54050****3.**H. R. Bennett and D. J. Lutzer,*Ordered space with -minimal base,*Topology Proceedings 2(1977), 371-382. MR**80j:54027****4.**-,*Problems in Perfect Ordered Spaces,*in: Open Problems in Topology, J. van Mill and G. M. Reed ed. (North-Holland, Amsterdam, 1990). CMP**91:03****5.**-,*A metric space of A. H. Stone and an example concerning -minimal bases,*Proc. Amer. Math. Soc.**126**(1998), 2191-2196. MR**98j:54054****6.**-,*Metrization, quasi-developments and -minimal bases*, Q. and A. in Gen. Top. 2(1984), 73-76. CMP**17:08****7.**R. Engleking, General Topology, (Hedermann, Berlin, 1989). MR**91c:54001****8.**I. Juhász, Cardinal Functions in Topology, (MC Tract 34, Mathematical Centre, Amsterdam, 1975). CMP**98:10****9.**D. J. Lutzer,*Twenty questions on ordered spaces,*in: Topology and Order Structures (Part 2), H. R. Bennett and D. J. Lutzer, editors. (MC Tract 169, Mathematical Centre, Amsterdam, 1983). MR**85h:54058****10.**S. Todor\v{c}evi\'{c},*Trees and linearly ordered sets*, in: Handbook of Set-Theoretic Topology, K. Kunen and J. E. Vanghan editors, (North-Holland, Amsterdam. 1984). MR**86h:54040**

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Additional Information

**Wei-Xue Shi**

Affiliation:
Department of Mathematics, Changchun Teachers College, Changchun 130032, China

Address at time of publication:
Institute of Mathematics, University of Tsukuba, Tsukuba-shi, Ibaraki 305, Japan

Email:
shi@abel.math.tsukuba.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-99-04853-4

Keywords:
\(\sigma\)-minimal base,
metrizable,
linearly ordered topological space,
special Aronszajn tree,
quasi-developable

Received by editor(s):
October 25, 1996

Received by editor(s) in revised form:
November 15, 1997

Published electronically:
April 15, 1999

Communicated by:
Alan Dow

Article copyright:
© Copyright 1999
American Mathematical Society