# Chris GoodUniversity of Birmingham · School of Mathematics

Chris Good

BA, MA, DPhil (Oxon)

## About

79

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Introduction

Chris is a mathematician at the University of Birmingham, UK. Chris's research areas are topology and discrete dynamical systems.

## Publications

Publications (79)

Assessment and feedback is an area where mathematical sciences departments have invested significant effort in recent times. Particular challenges have been identified relating to timely and detailed feedback, both of which are important given the widespread use of formative, and typically weekly, problem sheet assessments to aid and structure the...

Let f:X→X be a continuous map on a compact metric space X and let αf, ωf and ICTf denote the set of α-limit sets, ω-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map f has shadowing then every element of ICTf can be approximated (to any prescribed accuracy) by both the α-limit set and the ω-limit...

Let $f\colon X\to X$ be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map $f$ is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing and two-sided s-limit shadowing are eq...

We look at the preservation of various notions of shadowing in discrete dynamical systems under inverse limits, products, factor maps and the induced maps for symmetric products and hyperspaces. The shadowing properties we consider are the following: shadowing, h-shadowing, eventual shadowing, orbital shadowing, strong orbital shadowing, the first...

Continuous functions over compact Hausdorff spaces have been completely characterised. We consider the more general problem: given a set-valued function T from an arbitrary set X to itself, does there exist a compact Hausdorff topology on X with respect to which T is upper semicontinuous? We give conditions that are necessary for T to be upper semi...

We give a reformulation of the inverse shadowing property with respect to the class of all pseudo-orbits. This reformulation bears witness to the fact that the property is far stronger than might initially seem. We give some implications of this reformulation, in particular showing that systems with inverse shadowing are not sensitive. Finally we s...

Let $f\colon X\to X$ be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map $f$ is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing and two-sided s-limit shadowing are eq...

We study sensitivity, topological equicontinuity and even continuity in dynamical systems. In doing so we provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke dichotomy for minimal systems and show there exists a transitive system with an even continuit...

Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let X be a compact totally disconnected space and f:X→X a continuous map. We demonstrate that f has shadowing if and on...

Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and let $\alpha_f$, $\omega_f$ and $ICT_f$ denote the set of $\alpha$-limit sets, $\omega$-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map $f$ has shadowing then every element of $ICT_f$ can be approximated (to any prescrib...

We look at the preservation of various notions of shadowing in discrete dynamical systems under inverse limits, products, factor maps and the induced maps for symmetric products and hyperspaces. The shadowing properties we consider are the following: shadowing, h-shadowing, eventual shadowing, orbital shadowing, strong orbital shadowing, the first...

We give a reformulation of the inverse shadowing property with respect to the class of all pseudo-orbits. This reformulation bears witness to the fact that the property is far stronger than might initially seem. We give some implications of this reformulation, in particular showing that systems with inverse shadowing are not sensitive. Finally we s...

We discuss topological equicontinuity and even continuity in dynamical systems. In doing so we provide a classification of topologically transitive dynamical systems in terms of equicontinuity pairs, give a generalisation of the Auslander-Yorke Dichotomy for minimal systems and show there exists a transitive system with an even continuity pair but...

Let f be a unimodal map of the interval with critical point c. If the orbit of c is not dense then most points in lim{[0,1],f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim {{\omega}(c),f| {\om...

The complexity of biological models makes methods for their analysis and understanding highly desirable. Here, we demonstrate the orchestration of various novel coarse-graining methods by applying them to the mitotic spindle assembly checkpoint. We begin with a detailed fine-grained spatial model in which individual molecules are simulated moving a...

We study three notions of shadowing: classical shadowing, limit (or asymptotic) shadowing, and s-limit shadowing. We show that classical and s-limit shadowing coincide for tent maps and, more generally, for piecewise linear interval maps with constant slopes, and are further equivalent to the linking property introduced by Chen in 1991. We also con...

We consider various notions from the theory of dynamical systems from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. These Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definition stated in terms of a metric i...

Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let $X$ be a compact totally disconnected space and $f:X\to X$ a continuous map. We demonstrate that $f$ has shadowing...

Let $f:X\rightarrow X$ be a continuous map on a compact metric space, let ${\it\omega}_{f}$ be the collection of ${\it\omega}$ -limit sets of $f$ and let $\mathit{ICT}(f)$ be the collection of closed internally chain transitive subsets. Provided that $f$ has shadowing, it is known that the closure of ${\it\omega}_{f}$ in the Hausdorff metric coinci...

We consider several generalized metric properties and study the relation between a space X satisfying such property and its n-fold symmetric product satisfying the same property.

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give...

Given a nonempty compact metric space X and a continuous function f : X → X, we study shadowing and h-shadowing for the induced maps on hyperspaces, particularly in symmetric products, Fn (X), and the hyperspace 2X of compact subsets of X. We prove that f has shadowing [h-shadowing] if and only if 2f has shadowing [h-shadowing].

Transitivity and dense periodic points are two main ingredients of Devaney chaos. There are many stronger properties than these two main ingredients that have been studied as a shortcut to chaos. In this paper, we focus on two of these, locally everywhere onto and a strong dense periodicity property, and show the implication of these properties on...

Given a non-empty compact metric space X and a continuous function f: X → X, we study the dynamics of the induced maps on the hyperspace of non-empty compact subsets of X and on various other invariant subspaces thereof, in particular symmetric products. We show how some important dynamical properties transfer across induced systems. These amongst...

We look at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime per...

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.

We revisit van Dalen and Wattelʼs characterization of linearly ordered topological spaces in terms of nests of open sets and use this to give a topological characterization of ordinals. In particular we characterize ω1.

We say a space X with property PP is a universal space for orbit spectra of homeomorphisms with propertyPP provided that if Y is any space with property PP and the same cardinality as X and h:Y→Yh:Y→Y is any (auto)homeomorphism then there is a homeomorphism g:X→Xg:X→X such that the orbit equivalence classes for h and g are isomorphic. We construct...

Streszczenie
For a continuous map $f$ on a compact metric space $(X,d)$, a set $D\subset X$ is internally chain transitive if for every $x,y\in D$ and every $\delta>0$ there is a sequence of points $\langle x=x_0,x_1,\ldots,x_n=y\rangle$ such that $d(f(x_i),x_{i+1})< \delta$ for $0\leq i< n$. In this paper, we prove that for tent maps with periodic...

We address various notions of shadowing and expansivity for continuous maps
restricted to a proper subset of their domain. We prove new equivalences of
shadowing and expansive properties, we demonstrate under what conditions
certain expanding maps have shadowing, and generalize some known results in
this area. We also investigate the impact of our...

It is well known that \omega-limit sets are internally chain transitive and
have weak incompressibility; the converse is not generally true, in either
case. However, it has been shown that a set is weakly incompressible if and
only if it is an abstract \omega-limit set, and separately that in shifts of
finite type, a set is internally chain transit...

For a continuous map f on a compact metric space (X,d), a subset D of X is
internally chain transitive if for every x and y in D and every delta > 0 there
is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) <
delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally
chain transitive; in earlier work it wa...

Given a cardinal kappa <= c, a subset of the plane is said to be a kappa-point set if and only if it meets every line in precisely kappa many points. In response to a question of Cobb, we show that for all 2 <= kappa, lambda < c there exists a kappa-point set which is homeomorphic to a lambda-point set, and further, we also show that it is consiste...

Let X,Y be sets with quasiproximities X◃ and Y◃ (where A◃B is interpreted as “B is a neighborhood of A”). Let f,g:X→Y be a pair of functions such that whenever CY◃D, then f−1[C]X◃g−1[D]. We show that there is then a function h:X→Y such that whenever CY◃D, then f−1[C]X◃h−1[D], h−1[C]X◃h−1[D] and h−1[C]X◃g−1[D]. Since any function h that satisfies h−...

Let f be a unimodal map of the interval with critical point c. If the orbit of c is not dense then most points in lim ← {[0,1],f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, ℐ, of inhomogeneities is equal to lim ← ω(c),f| ω(c) ....

A set Λ is internally chain transitive if for any x,yΛ and >0 there is an -pseudo-orbit in Λ between x and y. In this paper we characterize all ω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point zX with ω(z)=Λ if and only if Λ is internally cha...

Given a map T: X -> X on a set X we examine under what conditions there is a separable metrizable or an hereditarily Lindelof or a Lindelof topology on X with respect to which T is a continuous map. For separable metrizable and hereditarily Lindelof, it turns out that there is such a topology precisely when the cardinality of X is no greater than c...

According to Mack a space is countably paracompact if and only if its product with [0,1] is δ-normal, i.e. any two disjoint closed sets, one of which is a regular Gδ-set, can be separated. In studying monotone versions of countable paracompactness, one is naturally led to consider various monotone versions of δ-normality. Such properties are the su...

Given a map T : X ! X on a set X we examine under what con- ditions there is a separable metrizable or an hereditarily Lindelof or a Lindelof topology on X with respect to which T is a continuous map. For separable metrizable and hereditarily Lindelof, it turns out that there is a such a topology precisely when the cardinality ofX is no greater tha...

We give two examples of tent maps with uncount-able (as it happens, post-critical) ω-limit sets, which have isolated points, with interesting structures. Such ω-limit sets must be of the form C ∪ R, where C is a Cantor set and R is a scattered set. Firstly, it is known that there is a restriction on the topological structure of countable ω-limit se...

This chapter presents an overview of several groups of open problems that are currently of interest to researchers associated with the Galway Topology Colloquium. Topics include set and function universals, countable para-compactness, abstract dynamical systems, and the embedding ordering within families of topological spaces. The chapter introduce...

We continue the study of properties related to monotone countable paracompactness, investigating various monotone versions of $\delta$-normality. We factorize monotone normality and stratifiability in terms of these weaker properties.

One possible natural monotone version of countable paracompactness, MCP, turns out to have some interesting properties. We investigate various other possible monotone versions of countable paracompactness and how they are related.

We consider the following problem: given a set X and a function T:X→X, does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we als...

Berkeley problems in mathematics (3rd edn.), edited by de Souza Paulo Ney and Silva Jorge-Nuno . Pp.591. £30·50. 2004. ISBN 0 387 00892 6 (Springer). - Volume 90 Issue 518 - Chris Good

In this paper we examine the structure of countable closed invariant sets under a dynamical system on a compact met- ric space. We are motivated by a desire to understand the possible structures of inhomogeneities in one-dimensional non-hyperbolic sets (inverse limits of flnite graphs) particularly when those inho- mogeneities form a countable set....

We demonstrate that the set of topologically distinct inverse limit spaces of tent maps with a Cantor set for its postcritical !-limit set has cardinality of the con- tinuum. The set of folding points (i.e. points at which the space is not homeomorphic to the product of a zero-dimensional set and an arc) of each of these spaces is also a Cantor set...

We show that, if an MCP (monotonically countably para- compact) space fails to be collectionwise Hausdorfi, then there is a mea- surable cardinal and that, if there are two measurable cardinals, then there is an MCP space that fails to be collectionwise Hausdorfi.

A base $\mathcal{B}$ for a space $X$ is said to be sharp if, whenever $x\in X$ and $(B_n)_{n\in\omega}$ is a sequence of pairwise distinct elements of $\mathcal{B}$ each containing $x$, the collection $\{\bigcap_{j\le n}B_j:n\in\omega\}$ is a local base at $x$. We answer questions raised by Alleche et al. and Arhangel$'$ski\u{\i} et al. by showing...

A topological space is said to be a Lindelöf space or is said to have the Lindelöf property if every open cover of X has a countable sub cover. The Lindelöf property was introduced by Alexandroff and Urysohn in 1929, with the term ëLindelöfí referring back to Lindelöf's result that any family of open subsets of Euclidean space has a countable sub-f...

A number of generalizations of metrizability have been defined or characterized in terms of g-functions. We study symmetricg-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.

A base B for a space X is said to be sharp if, whenever x is an element of X and (B-n)(nis an element ofomega) is a sequence of pairwise distinct element of B each containing x, the collection {boolean AND(jless than or equal ton) B-j: n is an element of omega} is a base at the point x. We answer questions raised by Alleche et al. and Arhangel'skii...

Assuming the existence of infinitely many measurable cardinals, a finite lattice is isomorphic to the interval between two T3 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 T3 topologies on a countable set.

There is a quasi-developable 2-manifold with a G δ -diagonal, which is not developable. Consis-tently, the example can be made to be countably metacompact. © 2001 Elsevier Science B.V. All rights reserved.

In this paper we prove that a completely regular pseudocompact space with a quasi-regular-G -diagonal is metrizable.

We show that a space is MCP (monotone countable paracompact) if and only if it has property (∗), introduced by Teng, Xia and Lin. The relationship between MCP and stratifiability is highlighted by a similar characterization of stratifiability. Using this result, we prove that MCP is preserved by both countably biquotient closed and peripherally cou...

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 characterisa-tion of stratifiability. We also provide new proofs of monotone insertion results by Nyikos and Pan, and Kubiak. We then investigate insertion theorems f...

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 β-spaces and, for q-spaces (hence first countable spac...

We provide new proofs for the classical insertion theorems of Dowker and Michael. The proofs are geometric in nature and highlight the connection with the preservation of normality in products. Both proofs follow directly from the Katÿetov- Tong insertion theorem and we also discuss a proof of this.

It is a well established fact that in Zermelo-Fraenkel set theory, Tychono's Theorem, the statement that the product of compact topological spaces is compact, is equivalent to the Axiom of Choice. On the other hand, Urysohn's Metrization Theorem, that every regular second countable space is metrizable, is provable from just the ZF axioms alone. A....

We study the structure of spaces admitting a continuous bijection to the space of all countable ordinals with its usual order topology. We relate regularity, zero-dimensionality and pseudonormality. We examine the effect of covering properties and ω1-compactness and show that locally compact examples have a particularly nice structure assuming MA +...

If f is an autohomeomorphism of some space X, then βf denotes its Stone-Čech extension to βX. For each n ⩽ ω, we give an example of a first countable, strongly zero-dimensional subparacompact X and a map f such that every point of X has an orbit of size n under f and βf has a fixed point. We give an example of a normal, zero-dimensional X such that...

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 Urysohn's Metrization Theorem is entirely effective, whilst recalling that some choice is required for Urysohn's Lemma. R is paracompact and ω1 may be paracompact but never metrizabl...

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 (...

Assuming }⁄, we construct flrst countable, locally compact examples of a Dowker space, an anti-Dowker space containing a Dowker space, and a countably paracompact space with Dowker square. We embed each of these into manifolds, which again satisfy the above properties.

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 (...

A subset G of a topological space is said to be a regular Gδ 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 Gδ, can be separated by disjoint open sets. Mack has shown that a space X is countably paracompact if a...

We prove that if there is a model of set-theory which contains no first count-able, locally compact, scattered, countably paracompact space X, whose Tychonoff square is a Dowker space, then there is an inner model which contains a measurable cardinal. In this paper we always take space to mean Hausdorff topological space. A space is normal if every...

Let : → A subset of is internally chain transitive if, for any > 0 and any ∈ there is an of points in from to It is well known that sets are internally chain transitive. The converse is in general not true. However, it has been shown that in shifts of finite type, a set is an set if and only if it is internally chain transitive. In this paper, we l...