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Conference Program, University Of Dayton

Summer Conference on Topology and Its Applications

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications.

Cohen Reals And The Sequential Order Of Groups, Alexander Shibakov

Summer Conference on Topology and Its Applications

We show that adding uncountably many Cohen reals to a model of diamond results in a model with no countable sequential group with an intermediate sequential order. The same model has an uncountable group of sequential order 2. We also discuss related questions.

On A Construction Of Some Class Of Metric Spaces, Dariusz Bugajewski

Summer Conference on Topology and Its Applications

In this talk we are going to describe Sz´az’s construction of some class of metric spaces. Most of all we will analyze topological properties of metric spaces obtained by using Sz´az’s construction. In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we will discuss the relation between Sz´az’s construction and the “linking construction”. A particular attention will be drawn to the “floor” metric, the analysis of which provides some interesting observations.

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.

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 …

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 …

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= …

Totally Geodesic Surfaces In Arithmetic Hyperbolic 3-Manifolds, Benjamin Linowitz, Jeffrey S. Meyer

Summer Conference on Topology and Its Applications

In this talk we will discuss some recent work on the problem of determining the extent to which the geometry of an arithmetic hyperbolic 3-manifold M is determined by the geometric genus spectrum of M (i.e., the set of isometry classes of finite area, properly immersed, totally geodesic surfaces of M, considered up to free homotopy). In particular, we will give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum and analyze the growth of the genera of minimal surfaces across commensurability classes. These results have applications to the study of …

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 Chogoshvili Homology Theory Of Continuous Maps Of Compact Spaces, Anzor Beridze, Vladimer Baladze

Summer Conference on Topology and Its Applications

In this paper an exact homology functor from the category Mor_C of continuous maps of compact Hausdorff spaces to the category LES of long exact sequences of abelian groups is defined (cf. [2], [3], [5]). This functor is an extension of the Hu homology theory, which is uniquely defined on the category of continuous maps of finite CW complexes and is constructed without the relative homology groups [9]. To define the given homology functor we use the Chogoshvili construction of projective homology theory [7], [8]. For each continuous map f:X → Y of compact spaces, using the notion of …

Some New Completeness Properties In Topological Spaces, Cetin Vural, Süleyman Önal

Summer Conference on Topology and Its Applications

One of the most widely known completeness property is the completeness of metric spaces and the other one being of a topological space in the sense of Cech. It is well known that a metrizable space X is completely metrizable if and only if X is Cech-complete. One of the generalisations of completeness of metric spaces is subcompactness. It has been established that, for metrizable spaces, subcompactness is equivalent to Cech-completeness. Also the concept of domain representability can be considered as a completeness property. In [1], Bennett and Lutzer proved that Cech-complete spaces are domain representable. They also proved, in …

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.

On Roitman's Principle For Box Products, Hector Alonso Barriga-Acosta

Summer Conference on Topology and Its Applications

One of the oldest problems in box products is if the countable box product of the convergent sequence is normal. It is known that consistenly (e.g., b=d, d=c) the answer is affirmative. A recent progress is due to Judy Roitman that states a combinatorial principle which also implies the normality and holds in many models.

Although the countable box product of the convergent sequence is normal in some models of b < d < c, Roitman asked what happen with her principle in this models. We answer that Roitman's principle is true in some models of b < d < c.

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 …

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 …

Pseudo-Contractibility, Felix Capulín, Leonardo Juarez-Villa, Fernando Orozco

Summer Conference on Topology and Its Applications

Let X, Y be topological spaces and let f, g:X→ Y be mappings, we say that f is pseudo-homotopic to g if there exist a continuum C, points a, b ∈ C and a mapping H:X ×C → Y such that H(x, a)=f(x) and H(x, b)=g(x) for each x ∈ X. The mapping H is called a pseudo-homotopy between f and g. A topological space X is said to be pseudo-contractible if the identity mapping is pseudo-homotopic to a constant mapping in X. i.e., if there exist a continuum C, points a, b ∈ C, x₀ ∈ X and …

On The Tightness And Long Directed Limits Of Free Topological Algebras, Gábor Lukács, Rafael Dahmen

Summer Conference on Topology and Its Applications

For a limit ordinal λ, let (A_α)_{α < λ} be a system of topological algebras (e.g., groups or vector spaces) with bonding maps that are embeddings of topological algebras, and put A = ∪_{α < λ} A_α. Let (A, T) and (A, A) denote the direct limit (colimit) of the system in the category of topological spaces and topological algebras, respectively. One always has T ⊇ A, but the inclusion may be strict; however, if the tightness of A is smaller than the cofinality of λ, then A=T.

In 1988, Tkachenko proved …

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.

A New Class Of Dendrites Having Unique Second Symmetric Product, David Maya, José G. Anaya, Fernando Orozco Zitli

Summer Conference on Topology and Its Applications

The second symmetric product of a continuum X, F₂(X), is the hyperspace consisting of all nonempty subsets of X having at most two points. A continuum X has unique hyperspace F₂(X) provided that each continuum Y satisfying that F₂(X) and F₂(Y) are homeomorphic must be homeomorphic to X. In this talk, a new class of dendrites having unique F₂(X) will be presented.

On Product Stability Of Asymptotic Property C, Gregory C. Bell, Andrzej Nagórko

Summer Conference on Topology and Its Applications

Asymptotic property C is a dimension-like large-scale invariant of metric spaces that is of interest when applied to spaces with infinite asymptotic dimension. It was first described by Dranishnikov, who based it on Haver's topological property C. Topological property C fails to be preserved by products in very striking ways and so a natural question that remained open for some 10+ years is whether asymptotic property C is preserved by products. Using a technique inspired by Rohm we show that asymptotic property C is preserved by direct products of metric spaces.

Entropy In Topological Groups, Part 2, Dikran Dikranjan

Summer Conference on Topology and Its Applications

Entropy was introduced first in thermodynamics and statistical mechanics, as well as information theory. In the last sixty years entropy made its way also in topology, ergodic theory, as well as other branches of mathematics as algebra, geometry and number theory where dynamical systems appear in one way or another.

Roughly speaking, entropy is a non-negative real number or infinity assigned to a "selfmap" T of a "space" X, where the "space" X can be a topological or uniform space, a measure space, an abstract or topological group (or vector space) or just a set. The "selfmap" T can be, …

Entropy In Topological Groups, Part 1, Dikran Dikranjan

Summer Conference on Topology and Its Applications

Entropy was introduced first in thermodynamics and statistical mechanics, as well as information theory. In the last sixty years entropy made its way also in topology, ergodic theory, as well as other branches of mathematics as algebra, geometry and number theory where dynamical systems appear in one way or another.

Roughly speaking, entropy is a non-negative real number or infinity assigned to a "selfmap" T of a "space" X, where the "space" X can be a topological or uniform space, a measure space, an abstract or topological group (or vector space) or just a set. The "selfmap" T can be, …

Some Applications Of The Point-Open Subbase Game, David Guerrero Sanchez

Summer Conference on Topology and Its Applications

Given a subbase S of a space X, the game PO(S,X) is defined for two players P and O who respectively pick, at the n-th move, a point xn 2 X and a set Un 2 S such that xn 2 Un . The game stops after the moves {xn, Un : n 2 !} have been made and the player P wins if the union of the Un’s equals X; otherwise O is the winner. Since PO(S,X) is an evident modification of the well-known point-open game PO(X), the primary line of research is to describe the relationship between PO(X) …

A Compact Minimal Space Whose Cartesian Square Is Not Minimal, Jan P. Boronski, Alex Clark, Piotr Oprocha

Summer Conference on Topology and Its Applications

A compact metric space X is called minimal if it admits a minimal homeomorphism; i.e. a homeomorphism h:X→ X such that the forward orbit {hⁿ(x):n=1, 2, ...} is dense in X, for every x ∈ X. In my talk I shall outline a construction of a family of 1-dimensional minimal spaces from "A compact minimal space Y such that its square YxY is not minimal" whose existence answer the following long standing problem in the negative.

Problem. Is minimality preserved under Cartesian product in the class of compact spaces?

Note that for the fixed point property this question …

Shift Maps And Their Variants On Inverse Limits With Set-Valued Functions, Judy Kennedy, Kazuhiro Kawamura, Van Nall, Goran Erceg

Summer Conference on Topology and Its Applications

We study inverse limits with set-valued functions using a pull-back construction and representing the space as an ordinary inverse limit space, which allows us to prove some known results and their extensions in a unified scheme. We also present a scheme to construct shift dynamics on the limit space and give some examples using the construction.

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.

Liouville Numbers And One-Sided Ergodic Hilbert Transformations, David Constantine, Joanna Furno

Summer Conference on Topology and Its Applications

We examine one-sided ergodic Hilbert transforms for irrational circle rotations and some mean-zero functions. Our approach uses continued fraction expansions to specify rotations by Liouville numbers for which the transformation has everywhere convergence or divergence.

Spaces With No S Or L Subspaces, Joan Hart, Kenneth Kunen

Summer Conference on Topology and Its Applications

We show it consistent for spaces X and Y to be both HS and HL even though their product X ×Y contains an S-space. Recall that an S-space is a T₃ space that is HS but not HL.

More generally, consider spaces that contain neither an S-space nor an L-space. We say a space is ESLC iff each of its subspaces is either both HS and HL or neither HS nor HL. The "C" in "ESLC" refers to HC; a space is HC iff each of its subspaces has the ccc (countable chain condition) (iff the space has no …

The Specification Property And Infinite Entropy For Certain Classes Of Linear Operators, James Kelly, Will Brian, Tim Tennant

Summer Conference on Topology and Its Applications

We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces.

It is known from the work of Bartoll, Martinínez-Giménez, Murillo-Arcila (2014), and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We also show that these forms of chaos imply …

Normal Images Of A Product And Countably Paracompact Condensation, Jila Niknejad

Summer Conference on Topology and Its Applications

In 1997, Buzjakova proved that for a pseudocompact Tychonoff space X and λ = | βX|⁺, X condenses onto a compact space if and only if X×(λ+1) condenses onto a normal space. This is a condensation form of Tamano's theorem. An interesting problem is to determine how much of Buzjakova's result will hold if "pseudocompact" is removed from the hypothesis.

In this talk, I am going to show for a Tychonoff space X, there is a cardinal λ such that if X×(λ+1) condenses onto a normal space, then X condenses onto a countably paracompact space.