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Reference: Jones, Mark R., (1993). Intersection topologies. DPhil. University of Oxford.Citable link to this page:


Intersection topologies

Abstract: Given two topologies, T1 and T2 on the same set X , the intersection topology with respectto T1 and T2 is the topology with basis{U1 ∩ U2 : U1 Є T1, U2 Є T2}Equivalently, T is the join of T1 and T2 in the lattice of topologies on the set X . Thisthesis is concerned with analysing some particular classes of intersection topologies, andalso with making some more general remarks about the technique.Reed was the first to study intersection topologies in these terms, and he made anextensive investigation of intersection topologies on a subset of the reals of cardinality N1, where the topologies under consideration are the inherited real-line topology and thetopology induced by an ω1-type ordering of the set. We consider the same underlyingset, and describe the properties of the intersection topology with respect to the inheritedSorgenfrey line topology and an ω1-type order topology, demonstrating that, whilst mostof the properties possessed by Reed's class are shared by ours, the two classes are strictlydisjoint.A useful characterisation of the intersection topology is as the diagonal of the productof the two topologies under consideration. We use this to prove some general propertiesabout intersection topologies, and also to show that the intersection topology withrespect to a first countable, hereditarily separable space and an ω1-type order topologycan never be locally compact.Results about the real line and Sorgenfrey line intersections with ω1 use various propertiesof the two lines. We demonstrate that most of the basic properties of the intersectiontopology require only the hereditary separability of R, and give examples to show that'hereditary' is essential here. We also show that results about normality, ω1-compactnessand the property of being perfect, all of which are set-theoretic in the classes of real-ω1 and Sorgenfrey-ω1 intersection topologies, can be shown to generalise to the class ofintersection topologies with respect to separable generalised ordered spaces and ω1.

Type of Award:DPhil Level of Award:Doctoral Awarding Institution: University of Oxford Notes:The digital copy of this thesis has been made available thanks to the generosity of Dr Leonard Polonsky


Reed, George M.More by this contributor



Dr. Mike ReedMore by this contributor


 Bibliographic Details

Issue Date: 1993Identifiers

Urn: uuid:292194e8-84c7-4c42-be33-a915b0e30067

Source identifier: 603853221 Item Description

Type: Thesis;

Language: eng Subjects: Topology Tiny URL: td:603853221


Author: Jones, Mark R. - institutionUniversity of Oxford facultyFaculty of Mathematical Sciences facultyMathematical and Physical Science



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