XII Galway Topology Colloquium 2008
Abstracts
Andy Barwell, University of Birmingham
For the interval I=[0, 1] and a tent map f:I→ I, the w-limit set of x ∈ I is the set of all limit points of the orbit of x. Here we present a characterization which relies heavily upon symbolic dynamics, and in so doing give some insight not only into the structure of w-limit sets but also of the symbol space of itineraries for the tent map.
Ben Chad, University of Oxford
Coauthors: Robin Knight, Rolf Suabedissen
A subset of the plane is said to be a two-point set iff it meets every line in exactly two points. We will discuss recent work which studies the isometry groups of two-point sets, and examine two consistency results which have arisen. In particular, we will show that it is consistent with ZFC that there exists a two-point set contained in a countable union of circles.
Gareth Davies, Mathematical Institute, Oxford University
On the surface, Lindelöfness seems as easy to work with as compactness. However, while there are many characterisations of compactness, it is not clear that the analogous formulations of Lindelöfness are equivalent. We consider one such formulation, that of being dually Lindelöf with respect to neighbourhood assignments. A space X is said to have this property if for every neighbourhood assignment {Ox | x ∈ X}, there is a Lindelöf subspace Y of X, such that {Oy | y ∈ Y} covers X. Among other things, we will see that this property is strictly weaker than Lindelöfness (a result by O. T. Alas, V. V. Tkachuk, and R. G. Wilson).
KP Hart, TU Delft
Coauthor: Alan Dow (UNCC)
I shall describe two compact connected Hausdorff spaces that are (consistently) not continuous images of the Cech-Stone remainder of the half line: one separable and one first-countable. The Continuum Hypothesis implies that all spaces from both classes are continuous images of the aforementioned remainder.
Michael Hrusak, Instituto de matematicas, UNAM, Mexico
Coauthors: J. Brendle, U. A. Ramos Garcia
We will discuss the problem of Malykhin asking whether there is a countable Fréchet topological group which is not metrizable.
Mercedes Jordan-Santana, CIMAT (Mexico)
Let C denote the complex plane and let I denote the unit interval. A geometric braid is a collection of n disjoint strands in C ×I where the k-th strand runs from (k, 0) to some point (h, 1), monotonically in the second variable and k, h ∈ {0, 1, ..., n}. Two geometric braids are equivalent if there exists an ambient isotopy, that fixes the end points of the braids and takes one braid to the other. The notation for this set is Bn. We define a multiplication between two braids by concatenation and with this operation we have that Bn is a group.
Singular braids SBn are defined in the same way as classical braids but we allow two strings to intersect transversally. The multiplication is also defined by concatenation but singular braids do not have an inverse, this is because applying an isotopy to singular braids does not undo the intersection points, and SBn is not a group. Nevertheless it is possible to create a group SGn which contains the singular braids: Fenn, Kenman and Rourke found a way to do so by coloring the singularities either in black or in white.
The main goal of this talk is to show that this group has no torsion. I will explain carefully all the definitions including: classical braid, singular braid, bands in a braid, etc. The proof of the free torsion will be combinatorial and very geometric.
Abdul Mohamad, Sultan Qaboos University
Coauthors: Paul Gartside & Bojana Pejić
A space X has property (a) iff for every open cover U of X and every dense subset D, there is a closed discrete subset F ⊆ D such that \st(F, \mcU)=X. We will discuss some questions about property (a) related to topological games, separable Moore spaces, manifolds and monotone normality.
Peter Nyikos, University of South Carolina
The celebrated Hahn-Mazurkiewicz theorem gave an internal characterization of continuous Hausdorff images of the closed unit interval: they are the compact, connected, locally connected, metrizable spaces. Such spaces are now known as Peano continua. Partly parallelling the well-known "metrization problem" was what might be called the "Peano problem": give an internal characterization of the images of compact, connected ordered spaces that is a natural generalization of the Hahn-Mazurkiewicz theorem.
It took almost a century after the publication of the Hahn-Mazurkiewicz theorem before this less-known problem was solved through the joint work of Jacek Nikiel and Mary Ellen Rudin. And until this year, very few people were aware of their elegant result:
THEOREM. A Hausdorff space is the continuous image of a compact, connected, ordered space if, and only if, it is compact, connected, locally connected, and monotonically normal.
This talk surveys the history of this theorem, and gives some recent related results and open problems.
Richard G. Wilson, Universidad Autónoma Metropolitana, Unidad Iztapalapa
Coauthors: Ofelia T. Alas, Vladimir V. Tkachuk
A neighbourhood assignment in a space X is a family O={Ox:x ∈ X} of open subsets of X such that x ∈ Ox for any x ∈ X. A set Y ⊆ X is a kernel of O if O(Y)=∪{Ox:x ∈ Y}=X. If every neighbourhood assignment in X has a closed and discrete (respectively, discrete) kernel, then X is said to be a D-space (respectively, a dually discrete space). We will discuss some new results concerning D-spaces and dually discrete spaces. Among other results, we show that every GO-space is dually discrete, every subparacompact scattered space and every continuous image of a Lindelöf P-space is a D-space and we prove an addition theorem for metalindelöf spaces which answers a question of Arhangel'skii and Buzyakova.
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