Weakened versions of normality in topological spaces

• Main Author: Calder, Christopher William
• Published: Queen's University Belfast 2002
• Subjects: 530.1
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.486046

The thesis first examines conditions on topological spaces - especially dense normality and kappa normality - that are known to be similar to but weaker than classical normality, and it extends the range of examples thaUllustrate them. Arising from this study, general methodologies are then developed and applied that permit the determination of whether a range of topological spaces - generated in specific ways - satisfy such conditions. Identification and axiomatising of a central concept, closely related to the classical notion of proximity and here called a 'separator', leads on to the study of topologies defined through separators and to the transfer of techniques from the initial specific examples to the general setting. It is established that these techniques are particularly effective when the underlying set carries a vector space structure; in particular, in this casea proof or disproof of dense normality - which is traditionally a relatively difficult problem - is easily obtained. Amore general view is derived of how to determine near-normality conditions for a wide variety of topological spaces. Amongst the by-products of the investigation is an iterative procedure of some independent interest (here termed 'echo sequence') that seeks to determine whether, in a given pair of disjoint closed sets, there are subsets on whkh the difficulty of separating them by neighbourhoods, as reqiJired by normality conditions, focuses. Results are obtained on the circumstances in which this focus of difficulty is empty: in which case, the determination of normality is simplified.