Nonmetrisable Manifolds

Abstract

Nyikos has defined a tree, denoted T, associated with any given Type I space. This thesis examines the properties of an T-tree if the space is a Type I nonmetrisable manifold. It is shown that a tree, T, is an T-tree of a Type I manifold iff T is a well-pruned ω1-tree. Furthermore, if T is any well-pruned ω1-tree, there are 2N1 different Type I manifolds for which T is the T-tree. The relationships between the properties of a Type I manifold and the properties of its T-tree are examined. It is shown that whenever a Type I manifold contains a copy of ω1, its T-tree must contain an uncountable branch. The thesis then addresses the problem of whether or not an arbitrary tree T admits a Type I manifold which is ω1-compact. If T does not contain an uncountable antichain, or a Suslin subtree, then there exists a Type I manifold with T-tree T. If T contains an uncountable antichain, then whether there exists an ω1-compact Type I manifold with T-tree T is undecidable. (*) implies there does not exist such a tree while ◊ implies that there does. If we assume ♣+, then at least one such manifold exists. ◊ also implies that if T contains a Suslin subtree, then there exists an ω1-compact manifold with T-tree T. Nyikos has recently defined a Type II space. We may associate an T-tree with such a space. This thesis shows that a Tree, T, admits a Type II manifold iff T has height not greater than ω1, and each level has cardinality no greater than c. The final chapter examines the relationship between microbundles and fibre bundles over nonmetrisable manifolds. In 1964 Milnor defined the notion of a microbunble. He ceased developing the theory of microbundles when later in the same year Kister showed that a microbundle over a metrisable manifold is equivalent to a fibre bundle. This thesis proves that the tangent microbundle over a manifold is a fibre bundle iff the manifold is metrisable. As a consequence of this we obtain further properties equivalent to metrisability in a manifold.