Topics in the General Topology of Non-metric Manifolds

Abstract

The main core of this thesis is the general topology of non-metrisable manifolds. Although our emphasis was studying non-metrisable topological manifolds, we did not hesitate to consider topological properties which are useful in studying manifolds in a more general context. This thesis consists of ve chapters : The rst chapter consists of two parts and in the rst part we provide a perspective of what has been done in the eld of non-metrisable topological manifolds and motivation for doing research in this eld from a set-theoretical point of view. In this overview our emphasis is on distinguishing some classical examples according to separation properties such as Hausdor ness, the Tychono property, normality and perfect normality. The second half of this introductory chapter is devoted to the classi cation of one dimensional topological manifolds. The second chapter is mainly about topological manifolds which do not satisfy the Hausdor property with the emphasis on the studying the neighbourhoods of singular points. In the third chapter we show that the Pr ufer surface is realcompact. Our emphasis in Chapter 4 is on the !-bounded property which lies between compactness and countable compactness. This property is equivalent to compactness in metric spaces. We also generalise the classical theorem which says every continuous function from the set of countable ordinals with order topology to the real line is constant on a tail of the set of countable ordinals. We study several generalisations of this classical theorem. Chapter 5 is closely related to the Nyikos bagpipe theorem. The proof of Nyikos Bagpipe theorem can be divided into two parts. The rst part has a more set-theoretic nature and the second part has a more geometric topology nature. The rst part of the proof of the Bagpipe theorem is closely related to exhausting the manifold with an increasing !1-sequence of open subspaces of the manifold. We study some topological spaces which have a -exhaustion for some ordinal number !1. In this chapter we also generalise the de nition of a long pipe in higher dimensions. iv