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

Fundamenta Mathematicae (1995)

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

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topJones, 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

top- [1] M. R. Jones, Sorgenfrey-${\omega}_{1}$ intersection topologies, preprint, 1993.
- [2] J. L. Kelley, General Topology, Springer, New York, 1975.
- [3] K. Kunen, On ordinal-metric intersection topologies, Topology Appl. 22 (1986), 315-319. Zbl0593.54001
- [4] A. J. Ostaszewski, A characterisation of compact, separable, ordered spaces, J. London Math. Soc. 7 (1974), 758-760. Zbl0278.54032
- [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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