Radiality and spokes: a structural theory of convergence

This thesis is a wide-ranging investigation of convergence properties in topological spaces, primarily Fréchet-Urysohn and radial spaces. The former are spaces such that every point in a closure of a subset is the limit of a sequence from within that set. The latter is a generalisation, defined by replacing 'sequence' with 'transfinite sequence'. Although not all spaces have these properties, they form a large enough class to encompass many important examples of spaces. These convergence properties can and should be studied locally and structurally. The first is achieved by removing the quantification over points in the definitions. For the second, we introduce the notion of spokes for points in topological spaces, which are sub-spaces for which the point has a descending neighbourhood base. In Chapter I, we introduce several convergence properties, and recall how they are connected and characterised by particular quotient maps. We also introduce p-character and quasi-isolation, to give our results full generality by not assuming the T1-axiom. Our main focus is the development of the theory of spokes in Chapter II. Here, we study how spokes can be used to approximate the neighbourhood base of a radial point and how (transfinite) sequences converge. We prove several characterisation theorems for radial and Fréchet-Urysohn points and their relationship with independently-based points, which are described through nests. We also use spokes in productivity problems and variants of the Fréchet-Urysohn property. In the final chapter, we demonstrate how properties of spokes manifest in different settings. For example, in compactifications of locally-compact spaces, spoke structures at the point-at-infinity reflect into the compact structure of the original space. Other examples are obtained by dualities, characterising radiality in ring spectra or Stone spaces algebraically. Such results justify using internal structures to investigate convergence properties and the author wishes to continue this line of investigation for the foreseeable future.