Ordered spaces with special bases

Harold Bennett; David Lutzer

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 3, page 289-299
  • ISSN: 0016-2736

Abstract

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We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a G δ -diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered space is equivalent to the existence of an OIF base and to the existence of a sharp base. We give examples showing that these are the best possible results.

How to cite

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Bennett, Harold, and Lutzer, David. "Ordered spaces with special bases." Fundamenta Mathematicae 158.3 (1998): 289-299. <http://eudml.org/doc/212316>.

@article{Bennett1998,
abstract = {We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a $G_δ$-diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered space is equivalent to the existence of an OIF base and to the existence of a sharp base. We give examples showing that these are the best possible results.},
author = {Bennett, Harold, Lutzer, David},
journal = {Fundamenta Mathematicae},
keywords = {point-countable base; weakly uniform base; ω-in-ω base; open-in-finite base; sharp base; metrizable space; quasi-developable space; linearly ordered space; generalized ordered space; -in- base},
language = {eng},
number = {3},
pages = {289-299},
title = {Ordered spaces with special bases},
url = {http://eudml.org/doc/212316},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Bennett, Harold
AU - Lutzer, David
TI - Ordered spaces with special bases
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 3
SP - 289
EP - 299
AB - We study the roles played by four special types of bases (weakly uniform bases, ω-in-ω bases, open-in-finite bases, and sharp bases) in the classes of linearly ordered and generalized ordered spaces. For example, we show that a generalized ordered space has a weakly uniform base if and only if it is quasi-developable and has a $G_δ$-diagonal, that a linearly ordered space has a point-countable base if and only if it is first-countable and has an ω-in-ω base, and that metrizability in a generalized ordered space is equivalent to the existence of an OIF base and to the existence of a sharp base. We give examples showing that these are the best possible results.
LA - eng
KW - point-countable base; weakly uniform base; ω-in-ω base; open-in-finite base; sharp base; metrizable space; quasi-developable space; linearly ordered space; generalized ordered space; -in- base
UR - http://eudml.org/doc/212316
ER -

References

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  1. [AJRS] A. Arkhangel'skiĭ, W. Just, E. Reznichenko and P. Szeptycki, Sharp bases and weakly uniform bases versus point countable bases, Topology Appl., to appear. 
  2. [BR] Z. Balogh and M. E. Rudin, Monotone normality, ibid. 47 (1992), 115-127. Zbl0769.54022
  3. [B] H. Bennett, On quasi-developable spaces, Gen. Topology Appl. 1 (1971), 253-262. Zbl0222.54037
  4. [B2] H. Bennett, Point-countability in linearly ordered spaces, Proc. Amer. Math. Soc. 28 (1971), 598-606. Zbl0197.19101
  5. [BLP] H. Bennett, D. Lutzer, and S. Purisch, On dense subspaces of generalized ordered spaces, Topology Appl., to appear. Zbl0942.54026
  6. [EL] R. Engelking and D. Lutzer, Paracompactness in ordered spaces, Fund. Math. 94 (1976), 49-58. Zbl0351.54014
  7. [G] G. Gruenhage, A note on the point-countable base question, Topology Appl. 44 (1992), 157-162. Zbl0776.54022
  8. [HL] R. Heath and W. Lindgren, Weakly uniform bases, Houston J. Math. 2 (1976), 85-90. Zbl0318.54032
  9. [L] D. Lutzer. On generalized ordered spaces, Dissertationes Math. 89 (1971). 

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