Mathematics Commons^™

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) both satisfy P, …

Fiber Strong Shape Theory For Topological Spaces, Ruslan Tsinaridze, Vladimer Baladze

Summer Conference on Topology and Its Applications

The purpose of this paper is the construction and investigation of fiber strong shape theory for compact metrizable spaces over a fixed base space B₀ , using the fiber versions of cotelescop, fibrant space and SSDR-map. In the paper obtained results containing the characterizations of fiber strong shape equivalences, based on the notion of double mapping cylinder over a fixed space B₀. Besides, in the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space B₀. Our approach is based on the method of Mardešić-Lisica and instead of …

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 …

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 …

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) …

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.

Entropy Of Induced Continuum Dendrite Homeomorphisms, Jennyffer Bohorquez, Alexander Arbieto

Summer Conference on Topology and Its Applications

Let f: D → D be a dendrite homeomorphism. Let C(D) denote the hyperspace of all nonempty connected compact subsets of D endowed with the Hausdorff metric. Let C(f):C(D) → C(D) be the induced continuum homeomorphism. In this talk we sketch the proof of the following result: If there exists a nonrecurrent branch point then the topological entropy of C(f) is ∞.

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, T₁, and Hausdorff by mixing in locally Hausdorff and the compound axioms sober + T₁ and sober + T_D. We show the latter compound condition characterizes hereditary sobriety, and that locally Hausdorff fits strictly between Hausdorff and sober + T₁. Classes of examples are constructed, in part to show the non-reversibility of key implications.

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 …

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.

Sequential Decreasing Strong Size Properties, Miguel A. Lara, Fernando Orozco, Felix Capulín

Summer Conference on Topology and Its Applications

Let X be a continuum. A topological property P is said to be a sequential decreasing strong size property provided that if μ is a strong size map for C_n(X), {t_n} is a sequence in the interval (t, 1) such that limt_n = t and each fiber μ^-1 (t_n) has the property P, then μ^-1 (t) has the property P. We show that the following properties are sequential decreasing strong size properties: be a Kelley continuum, indecomposability, local connectedness, continuum chainability and unicoherence.

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.

Lifting Homeomorphisms Of Cyclic Branched Covers Of The Sphere, Rebecca R. Winarski, Tyrone Ghaswala

Summer Conference on Topology and Its Applications

Birman and Hilden ask: given finite branched cover X over the 2-sphere, does every homeomorphism of the sphere lift to a homeomorphism of X? For covers of degree 2, the answer is yes, but the answer is sometimes yes and sometimes no for higher degree covers. In joint work with Ghaswala, we completely answer the question for cyclic branched covers. When the answer is yes, there is an embedding of the mapping class group of the sphere into a finite quotient of the mapping class group of X. In a family where the answer is no, we find a presentation …

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 …

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.

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 …