Topology: the Journey Into Separation Axioms
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Nearly Metacompact Spaces
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 98 (1999) 191–201 Nearly metacompact spaces Elise Grabner a;∗, Gary Grabner a, Jerry E. Vaughan b a Department Mathematics, Slippery Rock University, Slippery Rock, PA 16057, USA b Department Mathematics, University of North Carolina at Greensboro, Greensboro, NC 27412, USA Received 28 May 1997; received in revised form 30 December 1997 Abstract A space X is called nearly metacompact (meta-Lindelöf) provided that if U is an open cover of X then there is a dense set D ⊆ X and an open refinement V of U that is point-finite (point-countable) on D: We show that countably compact, nearly meta-Lindelöf T3-spaces are compact. That nearly metacompact radial spaces are meta-Lindelöf. Every space can be embedded as a closed subspace of a nearly metacompact space. We give an example of a countably compact, nearly meta-Lindelöf T2-space that is not compact and a nearly metacompact T2-space which is not irreducible. 1999 Elsevier Science B.V. All rights reserved. Keywords: Metacompact; Nearly metacompact; Irreducible; Countably compact; Radial AMS classification: Primary 54D20, Secondary 54A20; 54D30 A space X is called nearly metacompact (meta-Lindelöf) provided that if U is an open cover of X then there is a dense set D ⊆ X and an open refinement V of U such that Vx DfV 2 V: x 2 V g is finite (countable) for all x 2 D. The class of nearly metacompact spaces was introduced in [8] as a device for constructing a variety of interesting examples of non orthocompact spaces. -
1. Introduction in a Topological Space, the Closure Is Characterized by the Limits of the Ultrafilters
Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 08–37 THE ULTRAFILTER CLOSURE IN ZF GONC¸ALO GUTIERRES Abstract: It is well known that, in a topological space, the open sets can be characterized using filter convergence. In ZF (Zermelo-Fraenkel set theory without the Axiom of Choice), we cannot replace filters by ultrafilters. It is proven that the ultrafilter convergence determines the open sets for every topological space if and only if the Ultrafilter Theorem holds. More, we can also prove that the Ultrafilter Theorem is equivalent to the fact that uX = kX for every topological space X, where k is the usual Kuratowski Closure operator and u is the Ultrafilter Closure with uX (A) := {x ∈ X : (∃U ultrafilter in X)[U converges to x and A ∈U]}. However, it is possible to built a topological space X for which uX 6= kX , but the open sets are characterized by the ultrafilter convergence. To do so, it is proved that if every set has a free ultrafilter then the Axiom of Countable Choice holds for families of non-empty finite sets. It is also investigated under which set theoretic conditions the equality u = k is true in some subclasses of topological spaces, such as metric spaces, second countable T0-spaces or {R}. Keywords: Ultrafilter Theorem, Ultrafilter Closure. AMS Subject Classification (2000): 03E25, 54A20. 1. Introduction In a topological space, the closure is characterized by the limits of the ultrafilters. Although, in the absence of the Axiom of Choice, this is not a fact anymore. -
Lecture 12: Matroids 12.1 Matroids: Definition and Examples
IE 511: Integer Programming, Spring 2021 4 Mar, 2021 Lecture 12: Matroids Lecturer: Karthik Chandrasekaran Scribe: Karthik Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publi- cations. Matroids are structures that can be used to model the feasible set of several combinatorial opti- mization problems. In this lecture, we will define matroids, consider some examples, introduce the matroid optimization problem and see its generality through some concrete optimization problems that it formulates, and see a BIP formulation of the matroid optimization problem. We will sub- sequently see that the LP-relaxation of the BIP formulation has an integral optimal solution (via the concept of TDI that we learnt in the previous lecture). This will ultimately broaden the family of IPs that can be solved by simply solving the LP relaxation. 12.1 Matroids: Definition and Examples Definition 1. Let N be a finite set and I ⊆ 2N (recall that 2N denotes the collection of all subsets of N, i.e., 2N := fS : S ⊆ Ng). 1. The pair (N; I) is an independence system if (i) ; 2 I and (ii) If A 2 I and B ⊆ A, then B 2 I (i.e., the set I has the hereditary property{see Figure 12.1). The sets in I are called independent sets. Figure 12.1: Hereditary property of independent sets: If A is an independent set and B is a subset of A, then B should also be an independent set. 2. An independence system (N; I) is a matroid if (iii) for all A; B 2 I with jBj > jAj there exists e 2 B n A such that A [ feg 2 I. -
Chapter 7 Separation Properties
Chapter VII Separation Axioms 1. Introduction “Separation” refers here to whether or not objects like points or disjoint closed sets can be enclosed in disjoint open sets; “separation properties” have nothing to do with the idea of “separated sets” that appeared in our discussion of connectedness in Chapter 5 in spite of the similarity of terminology.. We have already met some simple separation properties of spaces: the XßX!"and X # (Hausdorff) properties. In this chapter, we look at these and others in more depth. As “more separation” is added to spaces, they generally become nicer and nicer especially when “separation” is combined with other properties. For example, we will see that “enough separation” and “a nice base” guarantees that a space is metrizable. “Separation axioms” translates the German term Trennungsaxiome used in the older literature. Therefore the standard separation axioms were historically named XXXX!"#$, , , , and X %, each stronger than its predecessors in the list. Once these were common terminology, another separation axiom was discovered to be useful and “interpolated” into the list: XÞ"" It turns out that the X spaces (also called $$## Tychonoff spaces) are an extremely well-behaved class of spaces with some very nice properties. 2. The Basics Definition 2.1 A topological space \ is called a 1) X! space if, whenever BÁC−\, there either exists an open set Y with B−Y, CÂY or there exists an open set ZC−ZBÂZwith , 2) X" space if, whenever BÁC−\, there exists an open set Ywith B−YßCÂZ and there exists an open set ZBÂYßC−Zwith 3) XBÁC−\Y# space (or, Hausdorff space) if, whenever , there exist disjoint open sets and Z\ in such that B−YC−Z and . -
Universality for and in Induced-Hereditary Graph Properties
Discussiones Mathematicae Graph Theory 33 (2013) 33–47 doi:10.7151/dmgt.1671 Dedicated to Mieczys law Borowiecki on his 70th birthday UNIVERSALITY FOR AND IN INDUCED-HEREDITARY GRAPH PROPERTIES Izak Broere Department of Mathematics and Applied Mathematics University of Pretoria e-mail: [email protected] and Johannes Heidema Department of Mathematical Sciences University of South Africa e-mail: [email protected] Abstract The well-known Rado graph R is universal in the set of all countable graphs , since every countable graph is an induced subgraph of R. We I study universality in and, using R, show the existence of 2ℵ0 pairwise non-isomorphic graphsI which are universal in and denumerably many other universal graphs in with prescribed attributes.I Then we contrast universality for and universalityI in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(2ℵ0 ) properties in the lattice K of induced-hereditary properties of which ≤ only at most 2ℵ0 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property. Keywords: countable graph, universal graph, induced-hereditary property. 2010 Mathematics Subject Classification: 05C63. 34 I.Broere and J. Heidema 1. Introduction and Motivation In this article a graph shall (with one illustrative exception) be simple, undirected, unlabelled, with a countable (i.e., finite or denumerably infinite) vertex set. For graph theoretical notions undefined here, we generally follow [14]. -
Testing Hereditary Properties of Sequences
Testing Hereditary Properties of Sequences Cody R. Freitag1, Eric Price2, and William J. Swartworth3 1 Department of Computer Science, UT Austin, Austin, TX, USA [email protected] 2 Department of Computer Science, UT Austin, Austin, TX, USA [email protected] 3 Department of Computer Science, UT Austin, Austin, TX, USA [email protected] Abstract A hereditary property of a sequence is one that is preserved when restricting to subsequences. We show that there exist hereditary properties of sequences that cannot be tested with sublinear queries, resolving an open question posed by Newman et al. [20]. This proof relies crucially on an infinite alphabet, however; for finite alphabets, we observe that any hereditary property can be tested with a constant number of queries. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity Keywords and phrases Property Testing Digital Object Identifier 10.4230/LIPIcs.CVIT.2016.23 1 Introduction Property testing is the problem of distinguishing objects x that satisfy a given property P from ones that are “far” from satisfying it in some distance measure [13], with constant (say, 2/3) success probability. The most basic questions in property testing are which properties can be tested with constant queries; which properties cannot be tested without reading almost the entire input x; and which properties lie in between. This paper considers property testing of sequences under the edit distance. We say a length n sequence x is -far from another (not necessarily length-n) sequence y if the edit distance is at least n. One of the key problems in property testing is testing if a sequence is 1 monotone; a long line of work (see [10, 5, 7, 8] and references therein) showed that Θ( log n) queries are necessary and sufficient. -
In a Topological Space (X, )
Separation Suppose that we have a sequence xn in a topological space (X; ). In order to consider the convergencef ofg such a sequence, we considerT the corre- sponding convergence result for a metric space in terms of open sets: namely: A sequence converges to a limit l X if every (open) neighbourhood containing l contains all but a finite number of2 points of the sequence. Consequently, we define convergence in (X; ) by: A sequence converges to a limit l X if everyT open set containing l contains all but a finite number of points of the sequence.2 However, this definition can have some unexpected consequences. If = φ, X , then every sequence in (X; ) converges, and every l X is a limitT of thef sequence.g T 2 Similarly, if we have a sequence in MATH 3402 which converges to an element of S11 (say) then every element of S11 is a limit of the sequence. If we want convergent sequences to have unique limits in X, then we need to impose further restrictions on the topology of X. Basicly the problem arises because there are distinct points x1 and x2 such that every open set containing x1 contains x2. Hence any sequence converging to x2 converges to x1 also. (In the examples above, this works both ways, but this is not necessary. If we consider the set a; b with the topology φ, a ;X , then every open set containing b contains a, butf notg vice-versa.) f f g g This means that x1 is a limit point of the set x2 , and this latter set is not closed. -
METRIC SPACES and SOME BASIC TOPOLOGY
Chapter 3 METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under- standing and ability to make use of properties of U U1. The next goal is to generalize our work to Un and, eventually, to study functions on Un. 3.1 Euclidean n-space The set Un is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. For completeness, we include the following De¿nition 3.1.1 Let S and T be sets. The Cartesian product of S and T , denoted by S T,is p q : p + S F q + T . The Cartesian product of any ¿nite number of sets S1 S2 SN , denoted by S1 S2 SN ,is j b ck p1 p2 pN : 1 j j + M F 1 n j n N " p j + S j . The object p1 p2pN is called an N-tuple. Our primary interest is going to be the case where each set is the set of real numbers. 73 74 CHAPTER 3. METRIC SPACES AND SOME BASIC TOPOLOGY De¿nition 3.1.2 Real n-space,denotedUn, is the set all ordered n-tuples of real numbers i.e., n U x1 x2 xn : x1 x2 xn + U . Un U U U U Thus, _ ^] `, the Cartesian product of with itself n times. nofthem Remark 3.1.3 From MAT108, recall the de¿nition of an ordered pair: a b a a b . def This de¿nition leads to the more familiar statement that a b c d if and only if a bandc d. -
Lecture 13 — February, 1 2014 1 Overview 2 Matroids
Advanced Graph Algorithms Jan-Apr 2014 Lecture 13 | February, 1 2014 Lecturer: Saket Saurabh Scribe: Sanjukta Roy 1 Overview In this lecture we learn what is Matroid, the connection between greedy algorithms and matroids. We also look at some examples of Matroids e.g., Linear Matroids, Graphic Matroids etc. 2 Matroids 2.1 A Greedy Approach Let G = (V, E) be a connected undirected graph and let w : E ! R≥0 be a weight function on the edges. For MWST Kruskal's so-called greedy algorithm works. Consider Maximum Weight Matching problem. 1 a b 3 3 d c 4 Application of the greedy algorithm gives (d,c) and (a,b). However, (d,c) and (a,b) do not form a matching of maximum weight. It is obviously not true that such a greedy approach would lead to an optimal solution for any combinatorial optimization problem. It turns out that the structures for which the greedy algorithm does lead to an optimal solution, are the matroids. 2.2 Matroids Definition 1. A pair M = (E, I), where E is a ground set and I is a family of subsets (called independent sets) of E, is a matroid if it satisfies the following conditions: (I1) φ 2 IorI = ø: 1 (I2) If A0 ⊆ A and A 2 I then A0 2 I. (I3) If A, B 2 I and jAj < jBj, then 9e 2 (B n A) such that A [feg 2 I: The axiom (I2) is also called the hereditary property and a pair M = (E, I) satisfying (I1) and (I2) is called hereditary family or set-family. -
Arxiv:Math/9810074V1 [Math.GN] 12 Oct 1998
Unification approach to the separation axioms between ∗ T0 and completely Hausdorff Francisco G. Arenas,† Julian Dontchev‡and Maria Luz Puertas§ September 19, 2021 Abstract The aim of this paper is to introduce a new weak separation axiom that generalizes the separation properties between T1 and completely Hausdorff. We call a topological space (X,τ) a Tκ,ξ-space if every compact subset of X with cardinality ≤ κ is ξ-closed, where ξ is a general closure operator. We concentrate our attention mostly on two new concepts: kd-spaces and T 1 -spaces. 3 1 Introduction The definitions of most (if not all) weak separation axioms are deceptively simple. However, the structure and the properties of those spaces are not always that easy to comprehend. In this paper we try to unify the separation axioms between T0 and completely Hausdorff by introducing the concept of Tκ,ξ-spaces. We call a topological space (X, τ) a Tκ,ξ-space arXiv:math/9810074v1 [math.GN] 12 Oct 1998 if every compact subset of X with cardinality ≤ κ is ξ-closed where ξ is a given closure operator. With different settings on κ and ξ we derive most of the well-known separation properties ‘in the semi-closed interval [T0, T3)’. We are going to consider not only Kuratowski closure operators but more general closure operators, such as the λ-closure operator [1] for ∗1991 Math. Subject Classification — Primary: 54A05, 54D10; Secondary: 54D30, 54H05. Key words and phrases — θ-closed, δ-closed, Urysohn, zero-open, λ-closed, weakly Hausdorff, T0, T1, com- pletely Hausdorff, kc-space, kd-space, Tκ,λ-space, anti-compact. -
Jumps in Speeds of Hereditary Properties in Finite Relational Languages
JUMPS IN SPEEDS OF HEREDITARY PROPERTIES IN FINITE RELATIONAL LANGUAGES MICHAEL C. LASKOWSKI AND CAROLINE A. TERRY Abstract. Given a finite relational language L, a hereditary L-property is a class of L-structures closed under isomorphism and substructure. The speed of H is the function which sends an integer n ≥ 1 to the number of distinct elements in H with underlying set f1; :::; ng. In this paper we give a description of many new jumps in the possible speeds of a hereditary L-property, where L is any finite relational language. In particular, we characterize the jumps in the polynomial and factorial ranges, and show they are essentially the same as in the case of graphs. The results in the factorial range are new for all examples requiring a language of arity greater than two, including the setting of hereditary properties of k-uniform hypergraphs for k > 2. Further, adapting an example of Balogh, Bollob´as,and Weinreich, we show that for all k ≥ 2, there are hereditary properties of k-uniform hypergraphs whose speeds oscillate between functions near the upper and lower bounds of the penultimate range, ruling out many natural functions as jumps in that range. Our theorems about the factorial range use model theoretic tools related to the notion of mutual algebricity. 1. introduction A hereditary graph property, H, is a class of graphs which is closed under isomorphisms and induced subgraphs. The speed of H is the function which sends a positive integer n to jHnj, where Hn is the set of elements of H with vertex set [n]. -
Epireflections in Topological Algebraic Structures
REFLECTIONS IN TOPOLOGICAL ALGEBRAIC STRUCTURES JULIO HERNANDEZ-ARZUSA´ AND SALVADOR HERNANDEZ´ Abstract. Let C be an epireflective category of Top and let rC be the epireflective functor associated with C. If A denotes a (semi)topological algebraic subcategory of Top, we study when rC (A) is an epireflective subcategory of A. We prove that this is always the case for semi-topological structures and we find some sufficient conditions for topological algebraic structures. We also study when the epireflective functor preserves products, subspaces and other properties. In particular, we solve an open question about the coincidence of epireflections proposed by Echi and Lazaar in [6, Question 1.6] and repeated in [7, Question 1.9]. Finally, we apply our results in different specific topological algebraic structures. 1. Introduction In this paper we deal with some applications of epireflective functors in the inves- tigation of topological algebraic structures. In particular, we are interested in the following question: Let C be an epireflective subcategory of Top (i.e., productive and hereditary, hence containing the 1-point space as the empty product), and let rC be the epireflective functor associated with C. If A denotes a (semi)topological varietal subcategory of Top (that is, a subcategory that is closed under products, subalgebras and homomorphic images), we study when rC(A) is an epireflective subcategory of A. arXiv:1704.01146v2 [math.GN] 20 Nov 2018 From a different viewpoint, this question has attracted the interest of many researchers recently (cf. [18, 21, 22, 24, 25, 26, 27, 30]) and, to some extent, our motivation for Date: November 21, 2018.