Applied Mathematics Commons^™

Classifying Matchbox Manifolds, Olga Lukina

Summer Conference on Topology and Its Applications

A matchbox manifold is a compact connected foliated space, locally homeomorphic to the product of a Euclidean disk and a Cantor set. Strange attractors in dynamical systems, and exceptional minimal sets of smooth foliations present examples of matchbox manifolds. Many actions of profinite groups on trees can be suspended to obtain matchbox manifolds, and similar examples arise in other contexts and in other parts of mathematics.

Thus there is a natural problem of classifying matchbox manifolds. The most tractable class of matchbox manifolds is the class of weak solenoids which are the inverse limits of finite-to-one coverings of closed manifolds. …

Quotients Of N-Fold Hyperspaces, Sergio Macías, Javier Camargo

Summer Conference on Topology and Its Applications

iven a continuum X and an integer n ≥ 2, let C_n(X) be the n-fold hyperspace of X consisting of all nonempty closed subsets of X with at most n components. We consider the quotient space Cⁿ₁(X)=C_n(X)/C₁(X) with the quotient topology. We prove several properties. For example: Cⁿ₁(X) is unicoherent; if X has the property of Kelley, Cⁿ₁(X) is contractible; dim(C_n(X))=dim(Cⁿ₁(X)); both Cⁿ₁([0, 1]) and Cⁿ₁(S¹) are Cantor manifolds; etc.

On Cohomological Dimensions Of Remainders Of Stone-Čech Compactifications, Vladimer Baladze

Summer Conference on Topology and Its Applications

In the paper the necessary and sufficient conditions are found under which a metrizable space has the Stone-Cech compactification whose remainder has the given cohomological dimensions (cf. [Sm], Problem I, p.332 and Problem II, p.334, and [A-N]).

In the paper [B] an outline of a generalization of Cech homology theory was given by replacing the set of all finite open coverings in the definition of Cech (co)homology group (Ĥⁿ_f(X, A;G)) Ĥ_n^f(X, A;G) (see [E-S], Ch.IX, p.237) by the set of all finite open families of border open coverings [Sm₁].

Following Y. Kodama …

Properties Of Weak Domain Representable Spaces, Joe Mashburn

Summer Conference on Topology and Its Applications

We will explore some of the basic properties of weak domain representable (wdr) spaces, including hereditary properties and properties of products. In particular, we will construct a Baire space that is not wdr, show that products of wdr spaces are wdr, and demonstrate that the factors of a product that is wdr need not themselves be wdr. We will also show that if X is a wdr space and Y ⊆ X such that |Y|=|X| then Y is wdr. We can declare a subset of a wdr space X to be open or to consist of isolated points without losing …

Enriched Topology And Asymmetry, Stephen Rodabaugh, Jeffrey T. Denniston, Austin Melton

Summer Conference on Topology and Its Applications

Mathematically modeling the question of how to satisfactorily compare, in many-valued ways, both bitstrings and the predicates which they might satisfy-a surprisingly intricate question when the conjunction of predicates need not be commutative-applies notions of enriched categories and enriched functors. Particularly relevant is the notion of a set enriched by a po-groupoid, which turns out to be a many-valued preordered set, along with enriched functors extended as to be "variable-basis". This positions us to model the above question by constructing the notion of topological systems enriched by many-valued preorders, systems whose associated extent spaces motivate the notion of topological spaces …

Topology And Order, Tom Richmond

Summer Conference on Topology and Its Applications

We will discuss topologies as orders, orders on sets of topologies, and topologies on ordered sets. More specifically, we will discuss Alexandroff topologies as quasiorders, the lattice of topologies on a finite set, and partially ordered topological spaces. Some topological properties of Alexandroff spaces are characterized in terms of their order. Complementation in the lattice of topologies on a set and in the lattice of convex topologies on a partially ordered set will be discussed.

On Di-Injective T0-Quasi-Metric Spaces, Collins Amburo Agyingi

Summer Conference on Topology and Its Applications

We prove that every q-hyperconvex T0-quasi-metric space (X, d) is di-injective without appealing to Zorn’s lemma. We also demonstrate that QX as constructed by Kemajou et al. and Q(X) (the space of all Katˇetov function pairs on X) are di-injective. Moreover we prove that di-injective T0-quasi-metric spaces do not contain proper essential extensions. Among other results, we state a number of ways in which the the di-injective envelope of a T0-quasi-metric space can be characterized.

Hausdorff Dimension Of Kuperberg Minimal Sets, Daniel Ingbretson

Summer Conference on Topology and Its Applications

The Seifert conjecture was answered negatively in 1994 by Kuperberg who constructed a smooth aperiodic flow on a three-manifold. This construction was later found to contain a minimal set with a complicated topology. The minimal set is embedded as a lamination by surfaces with a Cantor transversal of Lebesgue measure zero. In this talk we will discuss the pseudogroup dynamics on the transversal, the induced symbolic dynamics, and the Hausdorff dimension of the Cantor set.

Generic Approximation And Interpolation By Entire Functions Via Restriction Of The Values Of The Derivatives, Maxim R. Burke

Summer Conference on Topology and Its Applications

A theorem of Hoischen states that given a positive continuous function ε:Rⁿ→R, an unbounded sequence 0 ≤ c₁ ≤ c₂ ≤ ... and a closed discrete set T ⊆ Rⁿ, any C^∞ function g:Rⁿ→R can be approximated by an entire function f so that for k=0, 1, 2, ..., for all x ∈ Rⁿ such that |x| ≥ c_k, and for each multi-index α such that |α| ≤ k,

(a) |(D α f)(x)-(D α g)(x)| < ε(x);

(b) (D α f)(x)=(D α g)(x) if x …

Revelation Of Nano Topology In Cech Rough Closure Spaces, V. Antonysamy, Llellis Thivagar, Arockia Dasan

Summer Conference on Topology and Its Applications

The concept of Cech closure space was initiated and developed by E. Cech in 1966. Henceforth many more research scholars set their minds in this theory and developed it to a new height. Pawlak.Z derived and gave shape to Rough set theory in terms of approximation using equivalence relation known as indiscernibility relation. Further Lellis Thivagar enhanced rough set theory into a topology, called Nano Topology, which has at most five elements in it and he also extended this into multi granular nano topology. The purpose of this paper is to derive Nano topology in terms of Cech rough closure …

Topology And Experimental Distinguishability, Gabriele Carcassi, Christine A. Aidala, David J. Baker, Mark J. Greenfield

Summer Conference on Topology and Its Applications

In this talk we are going to formalize the relationship between topological spaces and the ability to distinguish objects experimentally, providing understanding and justification as to why topological spaces and continuous functions are pervasive tools in the physical sciences. The aim is to use these ideas as a stepping stone to give a more rigorous physical foundation to dynamical systems and, in particular, Hamiltonian dynamics.

We will first define an experimental observation as a statement that can be verified using an experimental procedure. We will show that observations are not closed under negation and countable conjunction, but are closed under …

Sequential Order Of Compact Scattered Spaces, Alan Dow

Summer Conference on Topology and Its Applications

A space is sequential if the closure of set can be obtained by iteratively adding limits of converging sequences. The sequential order of a space is a measure of how many iterations are required. A space is scattered if every non-empty set has a relative isolated point. It is not known if it is consistent that there is a countable (or finite) upper bound on the sequential order of a compact sequential space. We consider the properties of compact scattered spaces with infinite sequential order.

On The Lindelöf Σ-Property And Some Related Conclusions, Reynaldo Rojas-Hernandez, Fidel Casarrubias-Segura, Salvador Garcia-Ferreira

Summer Conference on Topology and Its Applications

We will present some known and some new results about Lindelöf Σ-spaces. We extend some classical results about the Lindelöf and the Lindelöf Σ-property in spaces C_p(X) for compact X to the case when X is a Lindelöf Σ-space. We also present some results about the Lindelöf Σ-property in Σ_s-products. A result of Tkachenko is generalized by showing that the bound w(X) ≤ nw(X)^Nag(X) holds for regular (not necessarily Tychonoff) spaces. Finally we present the solution for two question posed by V. V. Tkachuk about Eberlein and Corson compact spaces.

Order, Distance, Closure And Convergence: Reconciling Competing Fundamental Topological Concepts, Walter Tholen

Summer Conference on Topology and Its Applications

Already in Hausdorff’s 1914 book, often considered the cradle of general topology, one finds traces of a discussion on the relative strengths of the concepts mentioned in the title of this talk. In fact, one may argue that Hausdorff anticipated the basic ideas of how to unify these concepts, which were developed only later on by many mathematicians over the course of a century, as propagated in Hofmann, Seal & Tholen. Indeed, Hausdorff thought of ordering points by assigning to every pair of them a (truth) value, just as a metric assigns to them a number. More importantly, he also …

Virtual Seifert Surfaces And Slice Obstructions For Knots In Thickened Surfaces, Micah Chrisman, Hans U. Boden, Robin Gaudreau

Summer Conference on Topology and Its Applications

Here we introduce the notion of virtual Seifert surfaces. Virtual Seifert surfaces may be thought of as a generalization of Gauss diagrams of virtual knots to spanning surfaces of a knot. This device is then employed to extend the Tristram-Levine signature function to AC knots. Using the AC signature functions and Tuarev’s graded genus invariant, we determine the slice status of all 76 almost classical knots having at most six crossings. The slice obstructions for AC knots are then extended to all virtual knots via the parity projection map. This map, which is computable from a Gauss diagram, sends a …

On Quasi-Uniform Box Products, Hope Sabao, Olivier Olela Otafudu

Summer Conference on Topology and Its Applications

In this talk, we preset the quasi-uniform box product, a topology that is finer than the Tychonov product topology but coarser than the uniform box product.

We then present various notions of completeness of a quasi-uniform space that are preserved by their quasi-uniform box product using Cauchy filter pairs.

Uncountably Many Quasi-Isometry Classes Of Groups Of Type Fp, Ignat Soroko, Robert Kropholler, Ian Leary

Summer Conference on Topology and Its Applications

An interplay between algebra and topology goes in many ways. Given a space X, we can study its homology and homotopy groups. In the other direction, given a group G, we can form its Eilenberg-Maclane space K(G, 1). It is natural to wish that it is `small' in some sense. If K(G, 1) space has n-skeleton with finitely many cells, then G is said to have type F_n. Such groups act naturally on the cellular chain complex of the universal cover for K(G, 1), which has finitely generated free modules in all dimensions up to n. On the …

The Isbell-Hull Of An Asymmetrically Normed Space, Olivier Olela Otafudu, Jurie Conradie, Hans-Peter Künzi

Summer Conference on Topology and Its Applications

In this talk, we discuss an explicit method to define the linear structure of the Isbell-hull of an asymmetrically normed space.

On Continua With Regular Non-Abelian Self Covers, Mathew Timm

Summer Conference on Topology and Its Applications

We look at a planar 2-dimensional continuum X which satisfy the following:

Given any finite group G there is an |G|-fold regular self cover f:X → X with G as its group of deck transformations.

A Trace Formula For Foliated Flows (Working Paper), Jesús A. Álvarez López, Yuri A. Kordyukov, Eric Leichtnam

Summer Conference on Topology and Its Applications

The talk, based on work in progress, will be about our progress to show a trace formula for foliated flows on foliated spaces, which has been conjectured by V. Guillemin, and later by C. Deninger with more generality. It describes certain Leftchetz distribution of the foliated flow, acting on some version of the leafwise cohomology, in terms of local data at the closed orbits and fixed points.

Which Topological Groups Arise As Automorphism Groups Of Locally Finite Graphs?, Xiao Chang, Paul Gartside

Summer Conference on Topology and Its Applications

Let Γ be a graph which is countable and locally finite (every vertex has finite degree). Then the automorphism group of Γ, Aut(Γ), with the pointwise topology has a compact, zero dimensional open normal subgroup. We investigate whether the converse holds.

Topologically Homogeneous Continua, Isometrically Homogeneous Continua, And The Pseudo-Arc, Janusz Prajs

Summer Conference on Topology and Its Applications

We use accumulated knowledge on topologically homogeneous continua, and, in particular, on the pseudo-arc, to investigate the properties of isometrically homogeneous continua.

Rigidity And Nonrigidity Of Corona Algebras, Paul Mckenney, Alessandro Vignati

Summer Conference on Topology and Its Applications

Shelah proved in the 1970s that there is a model of ZFC in which every homeomorphism of the Cech-Stone remainder of the natural numbers is induced by a function on the natural numbers. More recently, Farah proved that in essentially the same model, every automorphism of the Calkin algebra on a separable Hilbert space must be induced by a linear operator on the Hilbert space. I will discuss a common generalization of these rigidity results to a certain class of C*-algebras called corona algebras. No prerequisites in C*-algebra will be assumed.

Balanced And Functionally Balanced P-Groups, Menachem Shlossberg

Summer Conference on Topology and Its Applications

In relation to Itzkowitz's problem, we show that a c-bounded P-group is balanced if and only if it is functionally balanced. We prove that for an arbitrary P-group, being functionally balanced is equivalent to being strongly functionally balanced. A special focus is given to the uniform free topological group defined over a uniform P-space. In particular, we show that this group is (functionally) balanced precisely when its subsets B_n, consisting of words of length at most n, are all (resp., functionally) balanced.

Compactly Supported Homeomorphisms As Long Direct Limits, Rafael Dahmen, Gábor Lukács

Summer Conference on Topology and Its Applications

Let λ be a limit ordinal and consider a directed system of topological groups (G_α)_{α < λ} with topological embeddings as bonding maps and its directed union G=∪_{α < λ}G_α. There are two natural topologies on G: one that makes G the direct limit (colimit) in the category of topological spaces and one which makes G the direct limit (colimit) in the category of topological groups.

For λ = ω it is known that these topologies almost never coincide (Yamasaki's Theorem).

In my talk last year, I introduced the Long Direct Limit Conjecture, stating that …

Uncountable Discrete Sets And Forcing, Akira Iwasa

Summer Conference on Topology and Its Applications

Suppose that a space X has no uncountable discrete subspace. We will discuss if forcing can create an uncountable discrete subspace of X.

Locally Compact Groups: Traditions And Trends, Karl Heinrich Hofmann, Wolfgang Herfort, Francesco G. Russo

Summer Conference on Topology and Its Applications

For a lecture in the Topology+Algebra and Analysis section, the subject of locally compact groups appears particularly fitting: Historically and currently as well, the structure and representation theory of locally compact groups draws its methods from each of theses three fields of mathematics. Nowadays one might justifiably add combinatorics and number theory as sources. The example of a study of a class of locally compact groups called “near abelian,” undertaken by W. Herfort, K. H. Hofmann, and F. G. Russo, may be used to illustrate the liaison of topological group theory with this different areas of interest. Concepts like the …

Disjoint Infinity Borel Functions, Daniel Hathaway

Summer Conference on Topology and Its Applications

Consider the statement that every uncountable set of reals can be surjected onto R by a Borel function. This is implied by the statement that every uncountable set of reals has a perfect subset. It is also implied by a new statement D which we will discuss: for each real a there is a Borel function f_a : RtoR and for each function g : RtoR there is a countable set G(g) of reals such that the following is true: for each a in R and for each function g : R to R, if f_a is disjoint …

Dense Subsets Of Function Spaces With No Non-Trivial Convergent Sequences, Vladimir V. Tkachuk

Summer Conference on Topology and Its Applications

We will show that a monolithic compact space X is not scattered if and only if C_p(X) has a dense subset without non-trivial convergent sequences. Besides, for any cardinal κ ≥ c, the space R^κ has a dense subspace without non-trivial convergent sequences. If X is an uncountable σ-compact space of countable weight, then any dense set Y ⊂ C_p(X) has a dense subspace without non-trivial convergent sequences. We also prove that for any countably compact sequential space X, if C_p(X) has a dense k-subspace, then X is scattered.

Compactness Via Adherence Dominators, Bhamini M. P. Nayar, Terrence A. Edwards, James E. Joseph, Myung H. Kwack

Summer Conference on Topology and Its Applications

This talk is based on a joint work by T. A. Edwards, J. E. Joseph, M. H. Kwack and B. M. P. Nayar that apperared in the Journal of Advanced studies in Topology, Vol. 5 (4), 2014), 8 - 15. B

An adherence dominator on a topological space X is a function π from the collection of filterbases on X to the family of closed subsets of X satisfying A(Ω) ⊆ π(Ω) where A(Ω) is the adherence of Ω. The notations π(Ω) and A(Ω) are used for the values of the functions π and A and π(Ω) =⋂_Ω π F= …