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-developments and 𝛿𝜃-bases*, J. London Math. Soc. (2)**9**(1974/75), 197–204. MR**0388334**, https://doi.org/10.1112/jlms/s2-9.2.197**2.**H. R. Bennett and E. S. Berney,*Spaces with 𝜎-minimal bases*, Proceedings of the 1977 Topology Conference (Louisiana State Univ., Baton Rouge, La., 1977), I, 1977, pp. 1–10 (1978). MR**540595****3.**H. R. Bennett and D. J. Lutzer,*Ordered spaces with 𝜎-minimal bases*, Proceedings of the 1977 Topology Conference (Louisiana State Univ., Baton Rouge, La., 1977), II, 1977, pp. 371–382 (1978). MR**540616****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.**Harold R. Bennett and David J. Lutzer,*A metric space of A. H. Stone and an example concerning 𝜎-minimal bases*, Proc. Amer. Math. Soc.**126**(1998), no. 7, 2191–2196. MR**1487358**, https://doi.org/10.1090/S0002-9939-98-04785-6**6.**-,*Metrization, quasi-developments and -minimal bases*, Q. and A. in Gen. Top. 2(1984), 73-76. CMP**17:08****7.**Ryszard Engelking,*General topology*, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR**1039321****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*, Topology and order structures, Part 2 (Amsterdam, 1981) Math. Centre Tracts, vol. 169, Math. Centrum, Amsterdam, 1983, pp. 1–18. MR**736688****10.**S. Todorčević,*Trees and linearly ordered sets*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 235–293. MR**776625**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
54F05,
54G20,
54E35

Retrieve articles in all journals with MSC (1991): 54F05, 54G20, 54E35

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