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

Jones, M.

Jones, M.

Fundamenta Mathematicae, Tome 146 (1995), p. 153-158 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $U_{1} \cap U_{2} : U_{1} \in T_{1}, U_{2} \in T_{2}$ . 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.

Publié le : 1995-01-01

Zbl 0812.54003

EUDML-ID : urn:eudml:doc:212058

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     author = {M. Jones},
     title = {Intersection topologies with respect to separable GO-spaces and the countable ordinals},
     journal = {Fundamenta Mathematicae},
     volume = {146},
     year = {1995},
     pages = {153-158},
     zbl = {0812.54003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv146i2p153bwm}
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Jones, M. Intersection topologies with respect to separable GO-spaces and the countable ordinals. Fundamenta Mathematicae, Tome 146 (1995) pp. 153-158. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv146i2p153bwm/

Bibliographie

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[00001] [2] J. L. Kelley, General Topology, Springer, New York, 1975.

[00002] [3] K. Kunen, On ordinal-metric intersection topologies, Topology Appl. 22 (1986), 315-319. | Zbl 0593.54001

[00003] [4] A. J. Ostaszewski, A characterisation of compact, separable, ordered spaces, J. London Math. Soc. 7 (1974), 758-760. | Zbl 0278.54032

[00004] [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. | Zbl 0602.54001