Paul Gartside

The tukey order on compact subsets of separable metric spaces

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map (a Tukey quotient) φ : P → Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients of each other are said to be Tukey equivalent. LetDc be the partially ordered set of Tukey equivalence classes of directed sets of size≤ c. It is shown thatDc contains an antichain of size 2, and so has size 2. The elements of the antichain are of the formK(M ), the set of compact subsets of a separable metrizable space M , ordered by inclusion. The order structure of such K(M )’s under Tukey quotients is investigated. Relative Tukey quotients are introduced. Applications are given to function spaces and to the complexity of weakly countably determined Banach spaces and Gul’ko compacta. §

The hierarchy of Borel universal sets

Abstract A Σ α 0 -subset U of the product X × Y is a Σ α 0 -universal set of X parametrised by Y if every Σ α 0 -set of X is of the form U y ={x: (x,y)∈U} , for some y ∈ Y . Let n ∈ ω and α ∈ ω 1 . If X is a compact space with a Σ n 0 -universal set parametrised by Y , then for all m ∈ ω , w ( X )⩽ nw ( Y ), hd ( X m )⩽ hd ( Y m ), hL ( X m )⩽ hL ( Y m ) and hc ( X m )⩽ hc ( Y m ). If X is a compact perfect space with a Σ α 0 -universal parametrised by Y , then w ( X )⩽ nw ( Y ). The statements “every compact monotonically normal space with a Σ α 0 -universal set parametrised by a second countable space is metrisable” and “every compact, first countable space with a Σ α 0 -universal set parametrised by a second countable space is metrisable” are undecidable in ZFC . Relevant examples are presented.

Parametrizing open universals

Abstract All spaces are assumed to be regular Hausdorff topological spaces. If X and Y are spaces, then an open set U in X × Y is an open universal set parametrized by Y if for each open set V of X , there is y ∈ Y such that V={x∈X: (x,y)∈U} . A space Y is said to parametrize W (κ) if Y parametrizes an open universal set of each space of weight less than or equal to κ . The following are the important results of this paper. If a metrizable space of weight κ parametrizes W (κ) , then κ has countable cofinality. If κ is a strong limit of countable cofinality, then there is a metrizable space of weight κ parametrizing W (κ) . It is consistent and independent that there is a cardinal κ of countable cofinality, but not a strong limit, and a metrizable space of weight κ parametrizing W (κ) . It is consistent and independent that a zero-dimensional, compact first countable space parametrizing itself (equivalently, parametrizing all spaces of the same or smaller weight) must be metrizable.

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.

The Tukey Order and Subsets of ω 1

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ϕ : P → Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by K(X)

Diversity of p-adic analytic manifolds

\mathcal {K}(X)

The complexity of the set of squares in the homeomorphism group of the circle

the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of K(S)

Relative separation properties

\mathcal {K}(S)

On the space of subgroups of a compact group II

corresponding to various subspaces S of ω1, their Tukey invariants, and hence the Tukey relations between them. It is shown that ωω is a strict Tukey quotient of Σ(ωω1)

Open universal sets

{\Sigma }(\omega ^{\omega _{1}})