Balanced And Functionally Balanced P-Groups, 2017 University of Udine

#### 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.

Topology And Experimental Distinguishability, 2017 University of Michigan - Ann Arbor

#### 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 ...

Generic Approximation And Interpolation By Entire Functions Via Restriction Of The Values Of The Derivatives, 2017 University of Prince Edward Island

#### 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:**R**^{n}→**R**, an unbounded sequence 0 ≤ c_{1} ≤ c_{2} ≤ ... and a closed discrete set T ⊆ **R**^{n}, any C^{∞} function g:**R**^{n}→**R** can be approximated by an entire function f so that for k=0, 1, 2, ..., for all x ∈ **R**^{n} 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 ∈ T.

We show that if C ⊆ **R**^{n ...}

On The Tightness And Long Directed Limits Of Free Topological Algebras, 2017 Technische Universitat Darmstadt

#### 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 that the free topological group F(X) is sequential ...

Relationships Between Hereditary Sobriety, Sobriety, Td, T1, And Locally Hausdorff, 2017 Youngstown State University

#### Relationships Between Hereditary Sobriety, Sobriety, Td, T1, And Locally Hausdorff, Stephen Rodabaugh, Jeffrey T. Denniston, Austin Melton, Jamal K. Tartir

*Summer Conference on Topology and Its Applications*

This work augments the standard relationships between sobriety, T_{1}, and Hausdorff by mixing in locally Hausdorff and the compound axioms sober + T_{1} and sober + T_{D}. We show the latter compound condition characterizes hereditary sobriety, and that locally Hausdorff fits strictly between Hausdorff and sober + T_{1}. Classes of examples are constructed, in part to show the non-reversibility of key implications.

Relationships Between Topological Properties Of X And Algebraic Properties Of Intermediate Rings A(X), 2017 California State University, Long Beach

#### Relationships Between Topological Properties Of X And Algebraic Properties Of Intermediate Rings A(X), Joshua Sack

*Summer Conference on Topology and Its Applications*

A topological property is a property invariant under homeomorphism, and an algebraic property of a ring is a property invariant under ring isomorphism. Let C(X) be the ring of real-valued continuous functions on a Tychonoff space X, let C^{*}(X) ⊆ C(X) be the subring of those functions that are bounded, and call a ring A(X) an *intermediate ring* if C^{*}(X) ⊆ A(X) ⊆ C(X). For a class Q of intermediate rings, an algebraic property P *describes* a topological property T among Q if for all A(X), B(Y) ∈ Q if A(X) and B(Y ...

Quotients Of N-Fold Hyperspaces, 2017 Universidad Autonoma del Estado de Mexico

#### 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 C^{n}_{1}(X)=C_{n}(X)/C_{1}(X) with the quotient topology. We prove several properties. For example: C^{n}_{1}(X) is unicoherent; if X has the property of Kelley, C^{n}_{1}(X) is contractible; dim(C_{n}(X))=dim(C^{n}_{1}(X)); both C^{n}_{1}([0, 1]) and C^{n}_{1}(S^{1}) are Cantor manifolds ...

Extension Theorems For Large Scale Spaces Via Neighbourhood Operators, 2017 University of Tennessee, Knoxville

#### Extension Theorems For Large Scale Spaces Via Neighbourhood Operators, Thomas Weighill, Jerzy Dydak

*Summer Conference on Topology and Its Applications*

Coarse geometry is the study of the large scale behaviour of spaces. The motivation for studying such behaviour comes mainly from index theory and geometric group theory. In this talk we introduce the notion of (hybrid) large scale normality for large scale spaces and prove analogues of Urysohn’s Lemma and the Tietze Extension Theorem for spaces with this property, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from ...

Domains And Probability Measures: A Topological Retrospective, 2017 Tulane University

#### Domains And Probability Measures: A Topological Retrospective, Michael Mislove

*Summer Conference on Topology and Its Applications*

Domain theory has seen success as a semantic model for high-level programming languages, having devised a range of constructs to support various effects that arise in programming. One of the most interesting - and problematic - is probabilistic choice, which traditionally has been modeled using a domain-theoretic rendering of sub-probability measures as valuations. In this talk, I will place the domain-theoretic approach in context, by showing how it relates to the more traditional approaches such as functional analysis and set theory. In particular, we show how the topologies that arise in the classic approaches relate to the domain-theoretic rendering. We also describe ...

Braid Group Actions On Rational Maps, 2017 American Mathematical Society

#### Braid Group Actions On Rational Maps, Eriko Hironaka, Sarah Koch

*Summer Conference on Topology and Its Applications*

Rational maps are maps from the Riemann sphere to itself that are defined by ratios of polynomials. A special type of rational map is the ones where the forward orbit of the critical points is finite. That is, under iteration, the critical points all eventually cycle in some periodic orbit. In the 1980s Thurston proved the surprising result that (except for a well-understood set of exceptions) when the post-critical set is finite the rational map is determined by the “combinatorics” of how the map behaves on the post-critical set. Recently, there has been interest in the question: what happens if ...

On The Lindelöf Σ-Property And Some Related Conclusions, 2017 Universidad Nacional Autonoma de Mexico

#### 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.

Aperiodic Colorings And Dynamics, 2017 Universidade de Santiago de Compostela

#### Aperiodic Colorings And Dynamics, Ramon Barral Lijo, Jesús A. Álvarez López

*Summer Conference on Topology and Its Applications*

A graph coloring is strongly aperiodic if every colored graph in its hull has no automorphisms. The talk will describe a method to define strongly aperiodic colorings on graphs with bounded degree. This also provides an optimal bound for the strongly distinguishing number of a graph. Then some applications to the theory of foliated spaces and to tilings will be discussed.

On Roitman's Principle For Box Products, 2017 Universidad Nacional Autonoma de Mexico

#### 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.

On Product Stability Of Asymptotic Property C, 2017 University of North Carolina at Greensboro

#### 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.

On Cohomological Dimensions Of Remainders Of Stone-Čech Compactifications, 2017 Batumi Shota Rustaveli State University

#### 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 (Ĥ^{n}_{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 ...

Entropy In Topological Groups, Part 2, 2017 University of Udine

#### 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 ...

Which Topological Groups Arise As Automorphism Groups Of Locally Finite Graphs?, 2017 University of Pittsburgh

#### 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, 2017 California State University, Sacramento

#### 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.

On Quasi-Uniform Box Products, 2017 North-West University (South Africa)

#### 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.

Enriched Topology And Asymmetry, 2017 Youngstown State University

#### 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 ...