A metric space of A. H. Stone and an example concerning 𝜎-minimal bases

Bennett, Harold; Lutzer, David

doi:10.1090/S0002-9939-98-04785-6

A metric space of A. H. Stone and an example concerning $\sigma$-minimal bases
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by Harold R. Bennett and David J. Lutzer PDF

Proc. Amer. Math. Soc. 126 (1998), 2191-2196 Request permission

Abstract:

In this paper we use a metric space $Y$ due to A. H. Stone and one of its completions $X$ to construct a linearly ordered topological space $E = E(Y,X)$ that is Čech complete, has a $\sigma$-closed-discrete dense subset, is perfect, hereditarily paracompact, first-countable, and has the property that each of its subspaces has a $\sigma$-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.

References

C. E. Aull, Quasi-developments and $\delta \theta$-bases, J. London Math. Soc. (2) 9 (1974/75), 197–204. MR 388334, DOI 10.1112/jlms/s2-9.2.197
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), Lecture Notes in Math., Vol. 378, Springer, Berlin, 1974, pp. 41–45. MR 0415575
Harold R. Bennett, A note on point-countability in linearly ordered spaces, Proc. Amer. Math. Soc. 28 (1971), 598–606. MR 275377, DOI 10.1090/S0002-9939-1971-0275377-2
H. R. Bennett and E. S. Berney, Spaces with $\sigma$-minimal bases, Topology Proc. 2 (1977), no. 1, 1–10 (1978). MR 540595
H. R. Bennett and D. J. Lutzer, Ordered spaces with $\sigma$-minimal bases, Topology Proc. 2 (1977), no. 2, 371–382 (1978). MR 540616
Jan van Mill and George M. Reed (eds.), Open problems in topology, North-Holland Publishing Co., Amsterdam, 1990. MR 1078636
Bennett, H., Lutzer, D., and Purisch, S., Dense metrizable subspaces of ordered spaces, to appear.
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
M. J. Faber, Metrizability in generalized ordered spaces, Mathematical Centre Tracts, No. 53, Mathematisch Centrum, Amsterdam, 1974. MR 0418053
H. Herrlich, Ordnungsfähigkeit total-diskontinuierlicher Räume, Math. Ann. 159 (1965), 77–80 (German). MR 182944, DOI 10.1007/BF01360281
D. J. Lutzer, On generalized ordered spaces, Dissertationes Math. (Rozprawy Mat.) 89 (1971), 32. MR 324668
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
R. Pol, A perfectly normal locally metrizable non-paracompact space, Fund. Math. 97 (1977), no. 1, 37–42. MR 464178, DOI 10.4064/fm-139-1-37-47
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
A. H. Stone, On $\sigma$-discreteness and Borel isomorphism, Amer. J. Math. 85 (1963), 655–666. MR 156789, DOI 10.2307/2373113
J. M. van Wouwe, GO-spaces and generalizations of metrizability, Mathematical Centre Tracts, vol. 104, Mathematisch Centrum, Amsterdam, 1979. MR 541832