A metric space of A. H. Stone

and an example concerning -minimal bases

Authors:
Harold R. Bennett and David J. Lutzer

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2191-2196

MSC (1991):
Primary 54F05, 54D18, 54D30, 54E35

DOI:
https://doi.org/10.1090/S0002-9939-98-04785-6

MathSciNet review:
1487358

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Abstract: In this paper we use a metric space due to A. H. Stone and one of its completions to construct a linearly ordered topological space that is \v{C}ech complete, has a -closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a -minimal base for its relative topology. However, is not metrizable and is not quasi-developable. The construction of 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.

**[A1]**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**[A2]**C. E. Aull,*Some properties involving base axioms and metrizability*, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Carnegie-Mellon Univ. and Univ. of Pittsburgh, Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Springer, Berlin, 1974, pp. 41–45. Lecture Notes in Math., Vol. 378. MR**0415575****[B]**Harold R. Bennett,*A note on point-countability in linearly ordered spaces*, Proc. Amer. Math. Soc.**28**(1971), 598–606. MR**0275377**, https://doi.org/10.1090/S0002-9939-1971-0275377-2**[BB]**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****[BL1]**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****[BL2]**Jan van Mill and George M. Reed (eds.),*Open problems in topology*, North-Holland Publishing Co., Amsterdam, 1990. MR**1078636****[BLP]**Bennett, H., Lutzer, D., and Purisch, S., Dense metrizable subspaces of ordered spaces, to appear.**[E]**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****[Fa]**M. J. Faber,*Metrizability in generalized ordered spaces*, Mathematisch Centrum, Amsterdam, 1974. Mathematical Centre Tracts, No. 53. MR**0418053****[H]**H. Herrlich,*Ordnungsfähigkeit total-diskontinuierlicher Räume*, Math. Ann.**159**(1965), 77–80 (German). MR**0182944**, https://doi.org/10.1007/BF01360281**[L]**D. J. Lutzer,*On generalized ordered spaces*, Dissertationes Math. Rozprawy Mat.**89**(1971), 32. MR**0324668****[L2]**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****[P1]**R. Pol,*A perfectly normal locally metrizable non-paracompact space*, Fund. Math.**97**(1977), no. 1, 37–42. MR**0464178****[P2]**R. Pol,*A non-paracompact space whose countable product is perfectly normal*, Comment. Math. Prace Mat.**20**(1977/78), no. 2, 435–437. MR**519381****[St]**A. H. Stone,*On 𝜎-discreteness and Borel isomorphism*, Amer. J. Math.**85**(1963), 655–666. MR**0156789**, https://doi.org/10.2307/2373113**[vW]**J. M. van Wouwe,*GO-spaces and generalizations of metrizability*, Mathematical Centre Tracts, vol. 104, Mathematisch Centrum, Amsterdam, 1979. MR**541832**

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Additional Information

**Harold R. Bennett**

Affiliation:
Department of Mathematics, Texas Tech University, Lubbock, Texas 79409

**David J. Lutzer**

Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187

DOI:
https://doi.org/10.1090/S0002-9939-98-04785-6

Keywords:
Linearly ordered space,
generalized ordered space,
\v Cech complete,
paracompact,
perfect space,
$\sigma $-minimal base,
metrization theory

Received by editor(s):
April 25, 1996

Received by editor(s) in revised form:
January 1, 1997

Communicated by:
Franklin D. Tall

Article copyright:
© Copyright 1998
American Mathematical Society