Boundary properties and construction techniques in general topology

Thesis

The aim of this thesis is twofold. First, we investigate spaces defined by asserting that their nowhere dense subsets have certain properties. Secondly, we develop some techniques for the construction of topological spaces.

We consider spaces where the nowhere dense sets are asserted to have some property P, calling such spaces boundary-P. We show that if there are no Lusin spaces then every compact boundary-metrizable space is metrizable. Boundary-separability is also studied and we show that if there are no L-spaces then every boundary-separable space is separable.

By adapting the absolute dimension function of Arhangel'skii, we define the new concept of cohesion. We show that every compact cohesive and every Hausdorff, sequential cohesive space is scattered. However, we construct regular, crowded spaces of all finite cohesions though there are no regular spaces of transfinite cohesion. We consider too the preservation of cohesion under various mappings and under the formation of products.

Turning to construction, we consider the class of compact monotonically normal spaces. It is well-known that it contains the class of spaces which are the continuous images of compact ordered spaces but it is still open as to whether they are actually distinct classes. Using Watson's resolutions, we give a method for constructing monotonically normal spaces. Though this also preserves continuous images of arcs, we show that it is because of a powerful result of Cornette rather than any trivial observation.

We also examine more closely monotone normality in images of compact ordered spaces using the Collins-Roscoe structuring mechanism. From this, we extract a strong instance of the mechanism, linear chain (F), which is held by all images of ordered compacta and all proto-metrizable spaces and implies Junnila's concept of utter normality.

Elementary submodels are an important tool in the construction of topological spaces. We develop a general method for applying them in varying circumstances and illustrate it by constructing three examples: Balogh's Q-set space, Rudin's normal but not collectionwise Hausdorff space and Balogh's small Dowker space.

Authors: Cairns, P

• Publication date: 1995
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: cohesion; analytic topology; elementary submodels; general topology; boundary properties
• Subjects: Analytic Topology or Topology