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Arxiv:Math/0412498V1
A semifilter approach to selection principles Lubomyr Zdomsky February 8, 2020 Abstract In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal g is a lower bound of the additivity number of the σ-ideal generated by Menger subspaces of the Baire space, and under u < g every subset X of the real line with the property Split(Λ, Λ) is Hurewicz, and thus it is consistent with ZFC that the property Split(Λ, Λ) is preserved by unions of less than b subsets of the real line. Introduction In this paper we shall present two directions of applications of semifilters in selection principles on topological spaces. First, we shall consider preservation by unions of the Menger property. Trying to describe the σ-compactness in terms of open covers, K.Menger intro- duced in [Me] the following property, called the Menger property: a topological space X is said to have this property if for every sequence (un)n∈ω of open covers of X there exists a sequence (vn)n∈ω such that each vn is a finite subfamily of un and the collection {∪vn : n ∈ ω} is a cover of X. The class of Menger topological spaces, i.e. spaces having the Menger property appeared to be much wider than the class of σ-compact spaces (see [BT], [CP], [JMSS] and many others), but it has interesting properties itself and poses a number of open questions. One of them, namely the question about the value of additivity of corresponding σ-ideal, arXiv:math/0412498v1 [math.GN] 27 Dec 2004 will be discussed in this paper. -
Arxiv:1603.03361V3 [Math.GN] 18 May 2016 Eeecs R O Eddfrtermidro Hspaper
PRODUCTS OF MENGER SPACES: A COMBINATORIAL APPROACH PIOTR SZEWCZAK AND BOAZ TSABAN Abstract. We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish pro- ductive properties for versions of Menger’s property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger’s property is strictly between Menger’s and Hurewicz’s classic properties. We include a number of open problems emerging from this study. 1. Introduction A topological space X is Menger if for each sequence U1, U2,... of open covers of the space X, there are finite subsets F1 ⊆ U1, F2 ⊆ U2, . whose union forms a cover of the space X. This property was introduced by Karl Menger [17], and reformulated as presented here by Witold Hurewicz [11]. Menger’s property is strictly between σ-compact and Lindelöf. Now a central notion in topology, it has applications in a number of branches of topology and set theory. The undefined notions in the following example, which are available in the indicated references, are not needed for the remainder of this paper. Example 1.1. Menger spaces form the most general class for which a positive solution of arXiv:1603.03361v3 [math.GN] 18 May 2016 the D-space problem is known [2, Corolarry 2.7], and the most general class for which a general form of Hindman’s Finite Sums Theorem holds [25]. -
HISTORICAL NOTES Chapter 1:. the Idea of Topologizing the Set of Continuous Functions from One Topological Space Into Another To
HISTORICAL NOTES Chapter 1:. The idea of topologizing the set of continuous functions from one topological space into another topological space arose from the notions of pointwise and uniform convergence of sequences of functions. Apparently the work of Ascoli [1883], [1889] and Hadamard [1898] marked the beginning of function space theory. The topology of pointwise convergence and the topology of uniform convergence are among the first function space topologies considered in the early years of general topology. The I supremum metric topology was studied in Frechet [19061. The paper of Tychonoff [1935] showed that the (Tychonoff) product on the set RX is nothing but the topology of pointwise convergence. In 1945, Fox [19451 defined the compact-open topology. Shortly thereafter, Arens [1946] studied this topology, which he called k-topology. Among other things which Arens proved was the compact-open topology version of Theorem 1.2.3. Set-open topologies in a more general setting were studied by Arens and Dugundji [1951] in connection with the concepts of admissible and proper topologies. Theorem 1.2.5 is due to Jackson [1952], and Example 1.2.7 can be found in Dugundji [1968]. Chapter 1:. Admissible [i.e., conjoining) topologies were introduced by Arens [1946] and splitting (i.e., proper) topologies were studied by Arens and Dugundji [1951], where they proved Theorem 2.5.3. Proofs of Theorem 2.5.2 and Corollary 2.5.4.a can be found in Fox [1945]. Corollary 2.5.7 is apparently due to Jackson [1952]; and Morita [1956] proved Corollary 2.5.8. -
Math 2150: Higher Arithmetic
MATH 2150: HIGHER ARITHMETIC Pamini Thangarajah Mount Royal University MATH 2150: Higher Arithmetic By Pamini Thangarajah, PhD. This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. The LibreTexts mission is to unite students, faculty and scholars in a cooperative effort to develop an easy-to-use online platform for the construction, customization, and dissemination of OER content to reduce the burdens of unreasonable textbook costs to our students and society. The LibreTexts project is a multi-institutional collaborative venture to develop the next generation of open-access texts to improve postsecondary education at all levels of higher learning by developing an Open Access Resource environment. The project currently consists of 13 independently operating and interconnected libraries that are constantly being optimized by students, faculty, and outside experts to supplant conventional paper-based books. These free textbook alternatives are organized within a central environment that is both vertically (from advance to basic level) and horizontally (across different fields) integrated. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. -
Vladimir V. Tkachuk Special Features of Function Spaces
Problem Books in Mathematics Vladimir V. Tkachuk A Cp-Theory Problem Book Special Features of Function Spaces Problem Books in Mathematics Series Editors: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA For further volumes: http://www.springer.com/series/714 Vladimir V. Tkachuk ACp-Theory Problem Book Special Features of Function Spaces 123 Vladimir V. Tkachuk Departamento de Matematicas Universidad Autonoma Metropolitana-Iztapalapa San Rafael Atlixco, Mexico City, Mexico ISSN 0941-3502 ISBN 978-3-319-04746-1 ISBN 978-3-319-04747-8 (eBook) DOI 10.1007/978-3-319-04747-8 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014933677 Mathematics Subject Classification (2010): 54C35 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. -
Selective Covering Properties of Product Spaces, II: Gamma Spaces
SELECTIVE COVERING PROPERTIES OF PRODUCT SPACES, II: γ SPACES ARNOLD W. MILLER, BOAZ TSABAN, AND LYUBOMYR ZDOMSKYY Abstract. We study productive properties of γ spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: (1) Solving a problem of F. Jordan, we show that for every unbounded tower set X ⊆ R of cardinality ℵ1, the space Cp(X) is productively Fr´echet–Urysohn. In particular, the set X is productively γ. (2) Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova– Scheepers, we prove that, assuming the Continuum Hypothesis, there are γ spaces whose product is not even Menger. (3) Solving a problem of Scheepers–Tall, we show that the properties γ and Gerlits–Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel’ski˘ı duality to obtain results concerning local properties of function spaces and countable topo- logical groups. 1. Introduction For a Tychonoff space X, let Cp(X) be the space of continuous real-valued functions on X, endowed with the topology of pointwise convergence, that is, the topology inherited from the Tychonoff product RX . In their seminal paper [14], Gerlits and Nagy characterized the property that the space Cp(X) is Fr´echet–Urysohn—that every point in the closure of a set arXiv:1310.8622v4 [math.LO] 10 Oct 2018 is the limit of a sequence of elements from that set—in terms of a covering property of the domain space X. -
Smarandache Sequences, Stereograms and Series
CHARLES ASHBACHER Smarandache Sequences, Stereograms and Series 1111111111111111111111111111111111111111 1111111111111111111111111111111111111111 1111122222222222222222222222222222211111 1111122222222222222222222222222222211111 1111122222333333333333333333332222211111 1111122222333333333333333333332222211111 1111122222333334444444444333332222211111 1111122222333334444444444333332222211111 1111122222333334444444444333332222211111 1111122222333334444444444333332222211111 1111122222333333333333333333332222211111 1111122222333333333333333333332222211111 1111122222222222222222222222222222211111 1111122222222222222222222222222222211111 1111111111111111111111111111111111111111 1111111111111111111111111111111111111111 Smarandache Stereogram Hexis Phoenix 2005 Charles Ashbacher Mount Mercy College Smarandache Sequences, Stereograms and Series Hexis Phoenix 2005 This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service) http://wwwlib.umi.com/bod/search/basic Peer Reviewers: Henry Ibstedt, Issy les Moulineaux, France Amarnath Murthy, Gujarat, India Lamarr Widmer, Messiah College, Grantham PA USA Copyright 2005 by Hexis and Charles Ashbacher Cover art © Kathleen Brogla and Charles Ashbacher Many books can be downloaded from the following E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1-931233-23-3 Standard Address Number: -
Spm Bulletin 35 (March 2013)
SPM BULLETIN ISSUE NUMBER 35: March 2013 CE 1. Editor’s note This issue contains a relatively large number of abstracts of papers dealing directly with selection principles. This is, in part, due to the stimulus created by the last SPM conference, followed by a special issue of Topology and its Applications that is under preparation. Pay special attention to Section 2.12 below, that announces a solution of the classic Malyhin’s Problem. This problem is closely related to the question of existence of γ- sets (one of the central themes of selection principles). In addition to its main result, this paper establishes that the existence of Malyhin groups need not imply that of γ-sets. Indeed, the result is quite flexible and may have additional consequences in the realm of SPM. With best regards, Boaz Tsaban, [email protected] http://www.cs.biu.ac.il/~tsaban 2. Long announcements 2.1. A characterization of the Menger property by means of ultrafilter con- vergence. We characterize various Menger-related properties by means of ultrafilter convergence, and discuss their behavior with respect to products. http://arxiv.org/abs/1210.2118 Paolo Lipparini arXiv:1303.4217v1 [math.GN] 18 Mar 2013 2.2. Topological spaces compact with respect to a set of filters. http://arxiv.org/abs/1210.2120 Paolo Lipparini If P is a family of filters over some set I, a topological space X is sequencewise P-compact if, for every I-indexed sequence of elements of X, there is F ∈ P such that the sequence has an F -limit point. -
Math Wonders to Inspire Teachers and Students
to Inspire Teachers and Students Alfred S. Posamentier to Inspire Teachers and Students Alfred S. Posamentier Association for Supervision and Curriculum Development Alexandria, Virginia USA Association for Supervision and Curriculum Development 1703 N. Beauregard St. * Alexandria, VA 22311-1714 USA Telephone: 800-933-2723 or 703-578-9600 * Fax: 703-575-5400 Web site: http://www.ascd.org * E-mail: [email protected] Gene R. Carter, Executive Director; Nancy Modrak, Director of Publishing; Julie Houtz, Director of Book Editing & Production; Darcie Russell, Project Manager; Technical Typesetting, Inc., Typesetting; Tracey Smith, Production Copyright © 2003 by Alfred S. Posamentier. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission from ASCD. Readers who wish to duplicate material may do so for a small fee by contacting the Copyright Clearance Center (CCC), 222 Rosewood Dr., Danvers, MA 01923, USA (telephone: 978-750-8400; fax: 978-750-4470; Web: http://www.copyright.com). ASCD has authorized the CCC to collect such fees on its behalf. Requests to reprint rather than photocopy should be directed to ASCD’s permissions office at 703-578-9600. Cover art copyright © 2003 by ASCD. Cover design by Shelley Young. ASCD publications present a variety of viewpoints. The views expressed or implied in this book should not be interpreted as official positions of the Association. All Web links in this book are correct as of the publication date below but may have become inactive or otherwise modified since that time. -
On the Definability of Menger Spaces Which Are Not/Sigma-Compact
On the definability of Menger spaces which are not σ-compact Franklin D. Tall1, Seçil Tokgöz2 July 19, 2016 Abstract Hurewicz proved completely metrizable Menger spaces are σ-compact. We extend this to Čech-complete Menger spaces, and consistently, to projective metrizable Menger spaces. On the other hand, it is consistent that there is a co-analytic Menger metrizable space that is not σ-compact. 1 When Menger spaces are σ-compact The Menger property is a strengthening of Lindelöfness which plays an important role in the study of Selection Principles. Definition 1.1. A space X is Menger if, given any sequence {Un}n<ω of arXiv:1607.04688v1 [math.GN] 16 Jul 2016 open covers of X, there exist finite subsets Vn of Un such that Sn<ω Vn covers X. 1Research supported by NSERC grant A-7354. 2The second author is supported by TÜBİTAK 2219 Grant. 2000 Math. Subj. Class. Primary 03E15, 03E35, 03E60, 54A25, 54D20, 54H05; Secondary 03E45, 54D40. Keywords and phrases: Menger, σ-compact, projective set of reals, analytic, co-analytic, K-analytic, determinacy, p-space, s-space, Σ-space. 1 Hurewicz [13] proved that completely metrizable Menger spaces are σ- compact and conjectured that indeed all Menger spaces are. His conjecture was disproved in [18]. Since then, easier, “natural” counterexamples have been constructed — see e.g. [32]. It was apparently not realized until now that Hurewicz’ theorem was not limited to metrizable spaces. We shall as- sume all spaces considered are completely regular. We shall prove: Theorem 1.2. Čech-complete Menger spaces are σ-compact. -
Arxiv:1911.04480V1 [Math.LO] 11 Nov 2019 Xssasequence a Exists Where Rnilscmsfrom Comes Principles X 03E05
SELECTION PRINCIPLES IN THE LAVER, MILLER, AND SACKS MODELS LYUBOMYR ZDOMSKYY Abstract. This article is devoted to the interplay between forcing with fusion and combinatorial covering properties. We discuss known in- stances of this interplay as well as present a new one, namely that in the Laver model for the consistency of the Borel’s conjecture, the Hurewicz property is preserved by finite products of metrizable spaces. 1. Introduction Combinatorial covering properties (or selection principles) arguably arose from the study of special sets of reals. These resolved many classical ques- tions in general topology and measure theory. As a result, information about special sets of reals is included in standard topology textbooks, such as Kuratowski’s Topology [28]. The most influential survey on special sets of reals is, probably, Miller’s chapter [35] in the Handbook of Set-Theoretic Topology. The most recent monograph on this topic is written by Bukovsky, see [14]. It complements nicely the classical book [7] of Bartoszynski and Judah. This theory still finds interesting applications in general topology, see, e.g., [23] for the interplay between λ-sets and homogeneity. A typical example of the evolution of special sets of reals into selection principles comes from strong measure zero sets (SMZ sets in what follows): X ⊂ R is SMZ if for every sequence hǫn : n ∈ ωi of positive reals there exists a sequence han : n ∈ ωi of “centers” such that X ⊂ Sn∈ω B(an, ǫn), where B(a, ǫ) = {x ∈ R : |x − a| < ǫ}. SMZ sets were introduced by Borel ca. a century ago who conjectured [13] that only countable sets have such property, i.e., that there are only trivial examples of SMZ sets. -
A Passion for Mathematics : Numbers, Puzzles, Madness, Religion, and the Quest for Reality / Clifford A
ffirs.qxd 5/13/05 11:21 AM Page iii AA PassionPassion forfor MathematicsMathematics Numbers, Puzzles, Madness, Religion, and the Quest for Reality CLIFFORD A. PICKOVER John Wiley & Sons, Inc. ffirs.qxd 5/13/05 11:21 AM Page vi ffirs.qxd 5/13/05 11:21 AM Page i AA PassionPassion forfor MathematicsMathematics ffirs.qxd 5/13/05 11:21 AM Page ii Works by Clifford A. Pickover The Alien IQ Test Black Holes: A Traveler’s Guide Calculus and Pizza Chaos and Fractals Chaos in Wonderland Computers, Pattern, Chaos, and Beauty Computers and the Imagination Cryptorunes: Codes and Secret Writing Dreaming the Future Egg Drop Soup Future Health Fractal Horizons: The Future Use of Fractals Frontiers of Scientific Visualization The Girl Who Gave Birth to Rabbits Keys to Infinity Liquid Earth The Lobotomy Club The Loom of God The Mathematics of Oz Mazes for the Mind: Computers and the Unexpected Mind-Bending Visual Puzzles (calendars and card sets) The Paradox of God and the Science of Omniscience The Pattern Book: Fractals, Art, and Nature The Science of Aliens Sex, Drugs, Einstein, and Elves Spider Legs (with Piers Anthony) Spiral Symmetry (with Istvan Hargittai) Strange Brains and Genius Sushi Never Sleeps The Stars of Heaven Surfing through Hyperspace Time: A Traveler’s Guide Visions of the Future Visualizing Biological Information Wonders of Numbers The Zen of Magic Squares, Circles, and Stars ffirs.qxd 5/13/05 11:21 AM Page iii AA PassionPassion forfor MathematicsMathematics Numbers, Puzzles, Madness, Religion, and the Quest for Reality CLIFFORD A. PICKOVER John Wiley & Sons, Inc.