Intersection topologies

Thesis

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.

Authors: Jones, M; Jones, Mark

• Publication date: 1993
• DOI: 10.5287/ora-wv5dxy6o4
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Topology