Aisling E. McCluskey

Representing set-inclusion by embeddability (among the subspaces of the real line)

Abstract We establish that the powerset P ( R ) of the real line R , ordered by set-inclusion, has the same ordertype as a certain subset of P ( R ) ordered by homeomorphic embeddability. This is a contribution to the ongoing study of the possible ordertypes of subfamilies of P ( R ) under embeddability, pioneered by Banach, Kuratowski and Sierpinski.

Representing quasi-orders by embeddability ordering of families of topological spaces

An elementary argument constructs, for each cardinal α, a topological space whose subspaces, ordered by homeomorphic embeddability, can model every partial order on α-many points. We show how to modify this procedure to deal also with quasi-orders (where the antisymmetry condition may fail), obtaining an initial estimate of the cardinality of the space then required.

Realizing Quasiordered Sets by Subspaces of ‘Continuum-Like’ Spaces

Given an ordered set E and a topological space X, we say that E can be realized within X if there is an injection j from E into the class of (homeomorphism classes of) subspaces of X such that, for x, y in E, x ≤ y if and only if j(x) is homeomorphically embeddable into j(y). It is known, for instance, that transfinite induction demonstrates that every partially-ordered set of cardinality c (and some larger ones) can be realized within the real line. We explore aspects of the realizability problem, indicating, in particular, how to weaken the hypothesis on E from partial- to quasi-order, and seeking to isolate the characteristics of the real line that are relevant here.

A Year of Engaging with the Discipline of Noticing: Five Mathematics Lecturers' Reflections.

In September 2010, five mathematics lecturers set out on a professional development project with the following aim: to reflect on teaching practice using John Masons Discipline of Noticing. At the end of the academic year, each lecturer considered her experiences of engaging with the process. In this paper, we describe the observations made and discuss the benefits and challenges of engaging with the Discipline of Noticing, namely, the benefits of a collaborative approach; the challenges of ‘noticing in the moment’ and the advantages of and difficulties with, writing brief-but-vivid accounts.

A Minimal Sober Topology Is Always Scott

Given the lattice of all topologies definable for an infinite set X, we employ he well‐developed contraction technique (of intersecting a given topology with a suitably chosen principal ultratopology) for solving many minimality problems. We confirm its potential in characterising and, where possible, identifying those topologies which are minimal with respect to certain separation axioms, notably that of sobriety and its conjunction with various other axioms. Finally, we offer an alternative description of each topologically established minimal structure in terms of the behavior of the naturally occurring specialization order and the intrin sic topology on the resulting partially‐ordered set.

The Minimal Structures for TA

Given the lattice of all topologies definable for an infinite set X, a technique to solve many minimality problems is developed. Its potential in characterizing and, where possible, identifying those topologies that are minimal with respect to various invariants, including TA, is illustrated. Finally, an alternative description of each topologically established minimal structure in terms of the behavior of the naturally occurring specialization order and the intrinsic topology on the resulting partially ordered set is offered.

Problems from the Galway Topology Colloquium

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 introduces the concept of universal as a space that in some sense parametrizes a collection of objects associated with a given topological space, such as the open subsets or the continuous real-valued functions. More precisely, the set or function universals can be defined as: Given a space X, a space Y parametrizes a continuous function universal i X via the function F if F : X × Y → ℝ is continuous and for any continuous f : X → ℝ there exists some y ∈ Y such that F ( x , y) = f ( x ) for all x ∈ X . The chapter describes problems regarding the construction of universals. Problems regarding the cardinal invariants of universals are also elaborated in the chapter.

The Propositional Theory of Closure

We study the simplest fragment of topological theory: those statements that can be expressed using one set variable, interior and closure operators, and inclusion. We introduce a formal system that is simple enough to be implemented on a computer and exhaustively studied and yet rich enough to be sound and complete for the fragment of theory under consideration. This fragment is rich enough to capture concepts such as regular open sets, extremal disconnectedness, partition topologies, and the nodec property.

An elementary counterexample on dense normality

Abstract We show that the Sorgenfrei plane is not normal on any of its dense subsets, that is, is not densely normal. This addresses in the simplest possible terms Arhangelskiis question as to whether an elementary example exists of a regular κ -normal space that fails to be densely normal.

a-7 – Comparison of Topologies (Minimal and Maximal Topologies)

Publisher Summary Inherent to the study of topology is the notion of comparison of two different topologies on the same underlying set. This comparison is carried out by the natural ordering of set inclusion. Thus, the family of all topologies definable for an infinite set X is a complete atomic and complemented lattice (under set inclusion) which is denoted as LT (X). If and S are two topologies on X with S ⊆ T, then S is said to be weaker or coarser than , and is said to be stronger or finer than S. Given a topological invariant P, a member of LT (X) is said to be minimal or maximal P if and only if (iff) possesses the property P. However, in the former case, no other weaker and in the later case no other stronger member of LT (X) possesses the property P. In the same way, P is said to be expansive or contractive iff for each P-member of LT (X), every stronger or in the later case every weaker member of LT (X) is also P.