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Title: Intersection topologies
Author: Jones, Mark R.
ISNI:       0000 0001 3592 5546
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 1993
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Given two topologies, T1 and T2 on the same set X , the intersection topology with respect to 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 . This thesis is concerned with analysing some particular classes of intersection topologies, and also with making some more general remarks about the technique. Reed was the first to study intersection topologies in these terms, and he made an extensive 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 the topology induced by an ω1-type ordering of the set. We consider the same underlying set, and describe the properties of the intersection topology with respect to the inherited Sorgenfrey line topology and an ω1-type order topology, demonstrating that, whilst most of the properties possessed by Reed's class are shared by ours, the two classes are strictly disjoint. A useful characterisation of the intersection topology is as the diagonal of the product of the two topologies under consideration. We use this to prove some general properties about intersection topologies, and also to show that the intersection topology with respect to a first countable, hereditarily separable space and an ω1-type order topology can never be locally compact. Results about the real line and Sorgenfrey line intersections with ω1 use various properties of the two lines. We demonstrate that most of the basic properties of the intersection topology 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-compactness and 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 of intersection topologies with respect to separable generalised ordered spaces and ω1.
Supervisor: Reed, George M. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Topology