David Gauld

Topological Properties of Manifolds

In this chapter we investigate the consequences for a manifold, when certain topological properties are assumed. In particular, we develop an important analytical tool, called partition of unity. 5.1 Compactness Recall that in a metric space X, a subset K is said to be compact, if every sequence from K has a subsequence which converges to a point in K. Recall also that every compact set is closed and bounded, and that the converse statement is valid for X = R n with the standard metric, that is, the compact subsets of R n are precisely the closed and bounded subsets. The generalization of compactness to an arbitrary topological space X does not invoke sequences. It originates from another important property of compact sets in a metric space, called the Heine-Borel property, which concerns coverings of K. Let X be a Hausdorff topological space, and let K ⊂ X.

Volterra spaces revisited

In this paper, we investigate Volterra spaces and relevant topological properties. New characterizations of weakly Volterra spaces are provided. An analogy of the Banach category theorem in terms of Volterra properties is obtained. It is shown that every weakly Volterra homogeneous space is Volterra, and there are metrizable Baire spaces whose hyperspaces of nonempty compact subsets endowed with the Vietoris topology are not weakly Volterra.

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.

The torsion of the group of homeomorphisms of powers of the long line

By blending techniques from set theory and algebraic topology we investigate the order of any homeomorphism of the n th power of the long ray or long line L having finite order, finding all possible orders when n = 1, 2, 3 or 4 in the first case and when n = 1 or 2 in the second. We also show that all finite powers of L are acyclic with respect to Alexander-Spanier cohomology.

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.

Homeomorphisms of 1‐Manifolds and ω‐Bounded 2‐Manifolds

ABSTRACT. A classification is given of continuous functions on the long line up to homotopy and homeomorphisms of the long line up to isotopy. The analogous problem for ω‐bounded 2‐manifolds is also investigated.

Non-metrisable manifolds

Topological Manifolds.- Edge of the World: When are Manifolds Metrisable?.- Geometric Tools.- Type I Manifolds and the Bagpipe Theorem.- Homeomorphisms and Dynamics on Non-Metrisable Manifolds.- Are Perfectly Normal Manifolds Metrisable?.- Smooth Manifolds.- Foliations on Non-Metrisable Manifolds.- Non-Hausdorff Manifolds and Foliations.

Enriched lattice-valued convergence groups

Considering a category SL-GConv, of stratified enriched cl-premonoid-valued generalized convergence spaces, we present the category SL-GConvGrp, of stratified L-generalized convergence groups, and some of its subcategories. We present two natural examples on stratified enriched cl-premonoid-valued convergence groups. Among other results, we show that every stratified strong L-limit group is SL-UCS-uniformizable.

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.

A strongly hereditarily separable, nonmetrisable manifold

Abstract Assuming the Continuum Hypothesis, a nonmetrisable manifold is constructed, all of whose finite powers are hereditarily separable.