Chris Good

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.

Continuing horrors of topology without choice

Abstract Various topological results are examined in models of Zermelo-Fraenkel set theory that do not satisfy the Axiom of Choice. In particular, it is shown that the proof of Urysohns Metrization Theorem is entirely effective, whilst recalling that some choice is required for Urysohns Lemma. R is paracompact and ω 1 may be paracompact but never metrizable. An example of a nonmetrizable paracompact manifold is given. Suslin lines, normality of LOTS and consequences of Countable Choice are also discussed.

Monotone insertion of continuous functions

Abstract We consider the problem of inserting continuous functions between pairs of semicontinuous functions in a monotone fashion. We answer a question of Pan and in the process provide a new characterization of stratifiability. We also provide new proofs of monotone insertion results by Nyikos and Pan, and Kubiak. We then investigate insertion theorems for hedgehog-valued functions providing monotone versions of two theorems due to Blair and Swardson. From this we provide new characterizations involving hedgehogs of monotonically normal spaces, stratifiable spaces, normal, countably paracompact spaces, and perfectly normal spaces. The proofs are mostly geometric in nature.

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.

On δ-normality

Abstract A subset G of a topological space is said to be a regular G δ if it is the intersection of the closures of a countable collection of open sets each of which contains G . A space is δ-normal if any two disjoint closed sets, of which one is a regular G δ , can be separated by disjoint open sets. Mack has shown that a space X is countably paracompact if and only if its product with the closed unit interval is δ-normal. Nyikos has asked whether δ-normal Moore spaces need be countably paracompact. We show that they need not. We also construct a δ-normal almost Dowker space and a δ-normal Moore space having twins.

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.

LARGE CARDINALS AND SMALL DOWKER SPACES

We prove that, if there is a model of set-theory which contains no flrst countable, locally compact, scattered Dowker spaces, then there is an inner model which contains a measurable cardinal. A Hausdorfi space is normal if, for every pair of disjoint closed sets C and D, there is a pair of disjoint open sets, U containing C and V containing D. A (normal) space is binormal if its product with the closed unit interval I is normal. It is fair to say that the study of normality, in particular the behaviour of normality in products and the difierence between normality and binormality, has played a central role in point-set topology. In 1951 Dowker (Do) introduced the notion of countable paracompactness, and proved that a normal space is binormal ifi it is countably paracompact. A space is (countably) paracompact if every (countable) open cover has a locally flnite open reflnement, however the important point to note, as far as we are concerned, is that countable paracompactness is the difierence between normality and binormal- ity. A quick study of Dowkers paper demonstrates just how natural the deflnition is|indeed, countable paracompactness is not so much a generalization of paracom- pactness, as one in a list of related properties which act to preserve normality-type conditions in products with a compact, metrizable factor: X £I is respectively or- thocompact, perfect, --normal, normal, perfectly or hereditarily normal, or mono- tonically normal ifi X is (respectively) countably metacompact (S), perfect (see (P 4.9)), countably paracompact (M), normal and countably paracompact (Do), per- fectly normal (Ka) and (P 4.9), monotonically normal and semi-stratiflable (G 5.22). Normal spaces that are not countably paracompact have become known as Dowker spaces, and it is natural to ask whether such spaces exist. In fact Dowker spaces do exist, but the example (Ru2), together with its mod- iflcations, has unsatisfyingly large cardinality and cardinal functions. This has prompted the generic deflnition of small Dowker spaces, i.e. ones of small size or small cardinal functions. Small Dowker spaces also exist, but, as yet, only with the help of various set-theoretic assumptions. For example: when | holds, there are

Symmetric g-functions

Abstract A number of generalizations of metrizability have been defined or characterized in terms of g -functions. We study symmetric g -functions which satisfy the condition that x ∈ g ( n , y ) iff y ∈ g ( n , x ). It turns out that the majority of symmetric g -functions fall into one of four known classes of space. Some metrization theorems are proved.

Measurable cardinals and finite intervals between regular topologies

Abstract Assuming the existence of infinitely many measurable cardinals, a finite lattice is isomorphic to the interval between two T 3 topologies on some set if and only if it is distributive. A characterisation is given for those finite lattices which are isomorphic to the interval between two T 3 topologies on a countable set.