Intersection topologies with respect to separable GO-spaces and the countable ordinals

M. Jones

Fundamenta Mathematicae (1995)

  • Volume: 146, Issue: 2, page 153-158
  • ISSN: 0016-2736

Abstract

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Given two topologies, T 1 and T 2 , on the same set X, the intersection topologywith respect to T 1 and T 2 is the topology with basis U 1 ∩ U 2 : U 1 ∈ T 1 , U 2 ∈ T 2 . Equivalently, T is the join of T 1 and T 2 in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and ω 1 -compactness in this class of topologies. We demonstrate that the majority of his results generalise to the intersection topology with respect to an arbitrary separable GO-space and ω 1 , employing a well-behaved second countable subtopology of the separable GO-space.

How to cite

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Jones, M.. "Intersection topologies with respect to separable GO-spaces and the countable ordinals." Fundamenta Mathematicae 146.2 (1995): 153-158. <http://eudml.org/doc/212058>.

@article{Jones1995,
abstract = {Given two topologies, $T_1$ and $T_2$, on the same set X, the intersection topologywith respect to $T_1$ and $T_2$ is the topology with basis $\{U_1 ∩ U_2 :U_1 ∈ T_1, U_2 ∈ T_2\}$. Equivalently, T is the join of $T_1$ and $T_2$ in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and $ω_1$-compactness in this class of topologies. We demonstrate that the majority of his results generalise to the intersection topology with respect to an arbitrary separable GO-space and $ω_1$, employing a well-behaved second countable subtopology of the separable GO-space. },
author = {Jones, M.},
journal = {Fundamenta Mathematicae},
keywords = {intersection topology; GO-space; separable; subtopology; normality; $ω_1$-compactness; countable ordinals; -compactness; separable GO-space},
language = {eng},
number = {2},
pages = {153-158},
title = {Intersection topologies with respect to separable GO-spaces and the countable ordinals},
url = {http://eudml.org/doc/212058},
volume = {146},
year = {1995},
}

TY - JOUR
AU - Jones, M.
TI - Intersection topologies with respect to separable GO-spaces and the countable ordinals
JO - Fundamenta Mathematicae
PY - 1995
VL - 146
IS - 2
SP - 153
EP - 158
AB - Given two topologies, $T_1$ and $T_2$, on the same set X, the intersection topologywith respect to $T_1$ and $T_2$ is the topology with basis ${U_1 ∩ U_2 :U_1 ∈ T_1, U_2 ∈ T_2}$. Equivalently, T is the join of $T_1$ and $T_2$ in the lattice of topologies on the set X. Following the work of Reed concerning intersection topologies with respect to the real line and the countable ordinals, Kunen made an extensive investigation of normality, perfectness and $ω_1$-compactness in this class of topologies. We demonstrate that the majority of his results generalise to the intersection topology with respect to an arbitrary separable GO-space and $ω_1$, employing a well-behaved second countable subtopology of the separable GO-space.
LA - eng
KW - intersection topology; GO-space; separable; subtopology; normality; $ω_1$-compactness; countable ordinals; -compactness; separable GO-space
UR - http://eudml.org/doc/212058
ER -

References

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  1. [1] M. R. Jones, Sorgenfrey- ω 1 intersection topologies, preprint, 1993. 
  2. [2] J. L. Kelley, General Topology, Springer, New York, 1975. 
  3. [3] K. Kunen, On ordinal-metric intersection topologies, Topology Appl. 22 (1986), 315-319. Zbl0593.54001
  4. [4] A. J. Ostaszewski, A characterisation of compact, separable, ordered spaces, J. London Math. Soc. 7 (1974), 758-760. Zbl0278.54032
  5. [5] G. M. Reed, The intersection topology with respect to the real line and the countable ordinals, Trans. Amer. Math. Soc. 297 (1986), 509-520. Zbl0602.54001

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