Sina Greenwood

On Volterra Spaces II

We say that a topological space X is Volterra if for each pair f, g: X→ℝ for which the sets of points at which f, respectively g, are continuous are dense, there is a common point of continuity; and X is strongly Volterra if in the same circumstances the set of common points of continuity is dense in X. For both of these concepts equivalent conditions are given and the situation involving more than two functions is explored.

Microbundles, manifolds and metrisability

The notion of a microbundle was introduced in the 1960s but the theory came to an abrupt halt when it was shown that for a metrisable manifold, microbundles are equivalent to fibre bundles. In this paper we consider microbundles over non-metrisable manifolds. In some cases microbundles are equivalent to fibre bundles but in others they are not. In particular, we show that a manifold is metrisable if and only if its tangent microbundle is equivalent to a fibre bundle. We also illustrate that for some non-metrisable manifolds every trivial microbundle contains a trivial fibre bundle whereas other manifolds may support a trivial microbundle not containing a trivial fibre bundle. 1. Definitions and notation Throughout this paper, by a manifold we mean a connected Hausdorff space in which each point has a neighbourhood homeomorphic to euclidean space. It is wellknown (cf. [4, p. 637]) that a manifold is metrisable if and only if it satisfies any one (and hence all) of the following properties: paracompact; σ-compact; second countable; meta-Lindelöf. In 1964 Milnor [3] introduced the notion of a microbundle as a means of transferring some of the procedures applicable to bundles over smooth manifolds to manifolds which are not necessarily smooth. The development came to an abrupt halt when Kister [2] showed that over metrisable manifolds every microbundle is equivalent to a fibre bundle. At the time the major effort in the study of manifolds was concentrated on compact manifolds and, as far as we know, the study of non-metrisable manifolds did not begin systematically until the late 1970s. Definition ([3]). A microbundle, denoted B i −→ E j −→ B, consists of topological spaces B and E, called the base space and the total space respectively, and continuous functions i and j, called the injection and projection maps respectively, such that the following conditions hold: • ji = 1B, the identity map on B; and • there is an open cover U of B so that for each U ∈ U there are a set V ⊂ j−1(U), with i(U) ⊂ V , and a homeomorphism φU : V −→ U × R Received by the editors July 8, 1997 and, in revised form, October 16, 1998. 2000 Mathematics Subject Classification. Primary 57N55, 54E35, 55R60, 57N05, 57N15.

Continuity in separable metrizable and Lindelöf spaces

First published in Proceedings of the American Mathematical Society 138(2):577-591 2010, published by the American Mathematical Society

HOMOGENEOUS AND INHOMOGENEOUS MANIFOLDS

All metaLindelof, and most countably paracompact, homogeneous manifolds are Hausdorff. Metacompact manifolds are never rigid. Every countable group can be realized as the group of autohomeomorphisms of a Lindelof manifold. There is a rigid foliation of the plane.

Constructing Type I nonmetrisable manifolds with given ϒ-trees☆

Abstract Nyikos has defined a tree, denoted ϒ , associated with any given Type I nonmetrisable manifold. In this paper, given an arbitrary well-pruned ω 1 -tree, T , we construct a Type I manifold such that its ϒ -tree is T . We show that whenever a Type I manifold contains a copy of ω 1 , its ϒ -tree must contain an uncountable branch. We address the problem of whether or not an arbitrary tree admits a Type I manifold which is ω 1 -compact.