Robin Knight

Monotone countable paracompactness

We study a monotone version of countable paracompactness, MCP, and of countable metacompactness, MCM. These properties are common generalizations of countable compactness and stratifiability and are shown to relate closely to the generalized metric g-functions of Hodel: MCM spaces coincide with -spaces and, for q-spaces (hence first countable spaces) MCP spaces coincide with wN-spaces. A number of obvious questions are answered, for example: there are “monotone Dowker spaces” (monotonically normal spaces that are not MCP); MCP, Moore spaces are metrizable; first countable (or locally compact or separable) MCP spaces are collectionwise Hausdorff (in fact we show that wN-spaces are collectionwise Hausdorff). The extent of an MCP space is shown to be no larger than the density and the stability of MCP and MCM under various topological operations is studied.

A characterization of ω-limit sets in shift spaces

A set 3 is internally chain transitive if for any x; y23 and > 0 there is an -pseudo-orbit in3 between x and y. In this paper we characterize all!-limit sets in shifts of finite type by showing that, if3 is a closed, strongly shift-invariant subset of a shift of finite type, X , then there is a point z2 X with!.z/D3 if and only if3 is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet B is the!-limit set of some point in the full shift space over B. We use similar techniques to prove that, for a tent map f , a closed, strongly f -invariant, internally chain transitive subset of the interval is the!-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space ZG (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the!-limit set of any point in ZG.

All Finite Distributive Lattices Occur as Intervals Between Hausdorff Topologies

It is shown that a finite lattice L is isomorphic to the interval betweentwo Hausdorff topologies on some set if and only if L is distributive. Thecorresponding results had previously been shown in ZFC for intervals between T1 topologies and, assuming the existence of infinitely manymeasurable cardinals, for intervals between T3 topologies.

Monotonically countably paracompact, collectionwise Hausdorff spaces and measurable cardinals

We show that, if an MCP (monotonically countably paracompact) space fails to be collectionwise Hausdorff, then there is a measurable cardinal and that, if there are two measurable cardinals, then there is an MCP space that fails to be collectionwise Hausdorff.

On the metrizability of spaces with a sharp base

Abstract A base B for a space X is said to be sharp if, whenever x ∈ X and ( B n ) n ∈ ω is a sequence of pairwise distinct element of B each containing x , the collection {⋂ j⩽n B j : n∈ω} is a base at the point x . We answer questions raised by Alleche et al. and Arhangelskii et al. by showing that a pseudocompact Tychonoff space with a sharp base need not be metrizable and that the product of a space with a sharp base and [0,1] need not have a sharp base. We prove various metrization theorems and provide a characterization along the lines of Ponomarevs for point countable bases.

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.

Quasi-developable manifolds

Abstract There is a quasi-developable 2-manifold with a G δ -diagonal, which is not developable. Consistently, the example can be made to be countably metacompact.

Finite Intervals in the Lattice of Topologies

We prove a conjecture of Reinhold: that a finite lattice is isomorphic to an interval in the lattice of topologies on some set if and only if it is isomorphic to an interval in the lattice of topologies on a finite set.