Countable Paracompactness and Weak Normality Properties
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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. -
Sequential Compactness Vs
SEQUENTIAL COMPACTNESS VS. COUNTABLE COMPACTNESS Angelo Bella and Peter Nyikos Abstract. The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal h is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindel¨of degree in most published models. Similar results, some definitive, are given for many other cardinal invariants. Special attention is paid to compact spaces. It is also shown that MA(!1) for σ-centered posets is equiv- alent to every countably compact T1 space with an !-in-countable base being second countable, and also to every compact T1 space with such a base being sequential. No separation axioms are assumed unless explicitly stated. 1. Introduction This article continues the theme, begun in [Ny], of sequential compactness (and lack thereof) in countably compact topological spaces, without the usual assumption of separation axioms. We do mention (and, in a few places, prove) some results involving the separation axioms T1, KC, Hausdorff (T2) and T3 (= T2 and regular) but we will always spell these axioms out when they are assumed. In [Ny] one of us gave some reasons for taking this unusual (for him) step. These will not be repeated here, but there is an additional reason, which is behind prac- tically all the results in this paper: quite unexpectedly, we have found countably compact spaces to be quite amenable to the techniques of modern set theory even in a general topological setting. -
Some Results on Separate and Joint Continuity
SOME RESULTS ON SEPARATE AND JOINT CONTINUITY A. BARECHE AND A. BOUZIAD Abstract. Let f : X × K → R be a separately continuous function and C a countable collection of subsets of K. Following a result of Calbrix and Troallic, there is a residual set of points x ∈ X such that f is jointly continuous at each point of {x}×Q, where Q is the set of y ∈ K for which the collection C includes a basis of neighborhoods in K. The particular case when the factor K is second countable was recently extended by Moors and Kenderov to any Cech-completeˇ Lindel¨of space K and Lindel¨of α-favorable X, improving a generalization of Namioka’s theorem obtained by Talagrand. Moors proved the same result when K is a Lindel¨of p-space and X is conditionally σ-α-favorable space. Here we add new results of this sort when the factor X is σC(X)-β-defavorable and when the assumption “base of neighborhoods” in Calbrix-Troallic’s result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces. 1. Introduction If K, X are topological spaces, a mapping f : X ×K → R is said to be separately continuous if for every x ∈ X and y ∈ K, the mappings f(x, .): K → R and f(., y): X → R are continuous, the reals being equipped with the usual topology. The spaces K and X satisfy the Namioka property N (X,K) if every separately continuous map f : X × K → R is (jointly) continuous at each point of a subset of X×K of the form R×K, where R is a dense subset of X [20]. -
On Countably Paracompact Spaces
ON COUNTABLY PARACOMPACT SPACES C. H. DOWKER LET X be a topological space, that is, a space with open sets such that the union of any collection of open sets is open and the intersection of any finite number of open sets is open. A covering of X is a collection of open sets whose union is X. The covering is called countable if it consists of a countable col lection of open sets or finite if it consists of a finite collection of open sets ; it is called locally finite if every point of X is contained in some open set which meets only a finite number of sets of the covering. A covering 53 is called a refinement of a covering U if every open set of 25 is contained in some open set of U. The space X is called countably paracompact if every countable covering has a locally finite refinement. The purpose of this paper is to study the properties of countably para compact spaces. The justification of the new concept is contained in Theorem 4 below, where it is shown that, for normal spaces, countable paracornpactness is equivalent to two other properties of known topological importance. 1. A space X is called compact if every covering has a finite refinement, paracompact if every covering has a locally finite refinement, and countably compact if every countable covering has a finite refinement. It is clear that every compact, paracompact or countably compact space is countably para compact. Just as one shows1 that every closed subset of a compact [para compact, countably compact] space is compact [paracompact, countably com pact], so one can show that every closed subset of a countably paracompact space is countably paracompact. -
On Pseudo-K-Spaces
Applied General Topology c Universidad Polit´ecnica de Valencia @ Volume 9, No. 2, 2008 pp. 177-184 On pseudo-k-spaces Annamaria Miranda Abstract. In this note a new class of topological spaces generalizing k-spaces, the pseudo-k-spaces, is introduced and investigated. Particu- lar attention is given to the study of products of such spaces, in analogy to what is already known about k-spaces and quasi-k-spaces. 2000 AMS Classification: 54D50, 54D99, 54B10, 54B15 Keywords: Quotient map, product space, locally compact space, (locally) pseudocompact space, pseudo-k-space. 1. Introduction The first example of two k-spaces whose cartesian product is not a k-space was given by Dowker (see [2]). So a natural question is when a k-space satisfies that its product with every k-spaces is also a k-space. In 1948 J.H.C. Whitehead proved that if X is a locally compact Hausdorff space then the cartesian product iX × g, where iX stands for the identity map on X, is a quotient map for every quotient map g. Using this result D.E. Cohen proved that if X is locally compact Hausdorff then X × Y is a k-space for every k-space Y (see Theorem 3.2 in [1]). Later the question was solved by Michael who showed that a k-space has this property iff it is a locally compact space (see [5]). A similar question, related to quasi-k-spaces, was answered by Sanchis (see [8]). Quasi-k-spaces were investigated by Nagata (see [7]) who showed that “a space X is a quasi-k-space (resp. -
Some Examples and Counterexamples of Advanced Compactness in Topology
International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 www.ijres.org Volume 9 Issue 1 ǁ 2021 ǁ PP. 10-16 Some Examples and Counterexamples of Advanced Compactness in Topology Pankaj Goswami Department of Mathematics, University of Gour Banga, Malda, 732102, West Bengal, India ABSTRACT: The examples and counter examples are always usefull for better comprehension of underlying concept in a theorem or definition .Compactness has come to be one of the most importent and useful topic in advanced mathematics.This paper is an attempt to fill in some of the information that the standard textbook treatment of compactness leaves out, and giving some constructive significative counterexamples of Advanced Compactness in Topology . KEYWORDS: open cover, compact, connected, topological space, example, counterexamples, continuous function, locally compact, countably compact, limit point compact, sequentially compact, lindelöf , paracompact --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 15-01-2021 Date of acceptance: 30-01-2021 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION We begin by presenting some definitions, notations,examples and some theorems that are essential for concept of compactness in advanced topology .For more detailed, see [1-2], [7-10] and others, in references .If X is a closed bounded subset of the real line ℝ, then any family of open sets in ℝ whose union contains X has a finite sub family whose union also contains X . If X is a metric space or topological space in its own right , then the above proposition can be thought as saying that any class of open sets in X whose union is equal to X has a finite subclass whose union is also equal to X. -
A Absolutely Countably Compact Space, 286 Accumulation Point, 81 of a Sequence of Sets, 80 Action, 218, 220, 222–228, 231
Index A B Absolutely countably compact space, 286 Baire property, 9, 69 Accumulation point, 81 Baire space, 9–11 of a sequence of sets, 80 Banach space, 80 Action, 218, 220, 222–228, 231, 234, 241, Base for the closed sets, 133 242 Base Tree Theorem, 274 Bernstein set, 158 algebraic, 220, 223, 234, 236 b f -group, 221, 224, 226–228 b f -action, 226 b f -space, 108, 122, 124, 125, 221, 222, 224, diagonal, 222 226–228, 231, 237 effective, 232 Birkhoff–Kakutani theorem, 39 faithful, 232 b-lattice, 164–167 free, 231 Bounded linearization of, 241 lattice, 164 trivial, 218, 229, 230, 240 neighborhood, 123 AD, see almost disjoint family open subset, 110 subset, 107–113, 115–119, 121–126, Additive group of the reals, 242 128–133, 136, 137, 139–142, 144–146, Admissible topology, 234 158, 221, 222 Alexandroff one-point compactification, 233 Boundedness, 111 Algebra of continuous functions, 221 Bounding number, 268 Almost compact space, 16, 17, 266 br -space, 108, 122, 124, 128, 129, 137 Almost disjoint family, 7, 8, 17, 28, 36, 195, 253–261, 266, 269, 271, 272, 280– 282, 284, 286 C Almost dominance with respect to a weak Canonical η-layered family, 263 selection, 280 Canonical open set in a GO-space, 170, 173 Almost metrizable topological group, 60 C-compact subset, 34, 36, 108, 115–117, Almost periodic point, 242 126–129, 137, 140, 184 ˇ Almost periodic subset, 242 Cech-complete space, 174–176, 181 Cech-completeˇ topological group, 181 Almost pseudo-ω-bounded space, 19, 20 Cellular family, 19 Almost P-space, 176 Cellularity, 49 (α, ) D -pseudocompact space, 77, 78, 85 C-embedded subset, 2–4, 15, 16, 18, 26, 110, Arcwise connected space, 184 122, 225 Arzelà-Ascoli theorem, 108, 239, 240 C∗-embedded subset, 2, 13, 17, 24, 25 Ascoli theorem, 80 Centered family, 275 © Springer International Publishing AG, part of Springer Nature 2018 291 M. -
Products of Countably Compact Spaces
PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 58, July 1976 PRODUCTS OF COUNTABLY COMPACT SPACES THOMAS W. RISHEL Abstract. Extensions of sufficient conditions for the product of two countably compact spaces to be countably compact, plus a relevant example. In 1953 Novak [4] published an example to show that countable compact- ness is not preserved under products. Novak's example consists of taking two countably compact subspaces Ax and A2 of the Stone-Cech compactification ßN of the natural numbers A such that Ax U A2 = ßN and Ax n A2 = A; the product Ax X A2 is not countably compact because it contains an infinite closed discrete space. Additional conditions are thus necessary on one of the countably compact spaces A or F to ensure countable compactness of the product A x F. Some of the additional properties on A which will guarantee this are: sequentially compact, first countable, sequential, k. These properties and some proofs have been discussed in a paper of S. Franklin [2]. Other properties which generate countably compact products in this manner are paracompactness and meta- compactness, since either of these conditions, when added to countable compactness of a factor, makes the factor compact, and the product of a compact space with a countably compact space is well known to be countably compact. In what follows, assume all spaces Hausdorff. A space which is a generalization of /:-space (hence, of first countable and sequential space) has proved fruitful in some product theorems. This space is called a weakly-^ space. Definition. A topological space A is weakly-/: iff a subset F of A is closed in X if F fi C is finite for every compact C in A. -
On Feebly Compact Paratopological Groups 11
ON FEEBLY COMPACT PARATOPOLOGICAL GROUPS TARAS BANAKH AND ALEX RAVSKY Abstract. We obtain many results and solve some problems about feebly compact paratopo- logical groups. We obtain necessary and sufficient conditions for such a group to be topologi- cal. One of them is the quasiregularity. We prove that each 2-pseudocompact paratopological group is feebly compact and that each Hausdorff σ-compact feebly compact paratopological group is a compact topological group. Our particular attention concerns periodic and topo- logically periodic groups. We construct examples of various compact-like paratopological groups which are not topological groups, among them a T0 sequentially compact group, a T1 2-pseudocompact group, a functionally Hausdorff countably compact group (under the axiomatic assumption that there is an infinite torsion-free Abelian countably compact topo- logical group without non-trivial convergent sequences), and a functionally Hausdorff second countable group sequentially pracompact group. We prove that the product of a family of feebly compact paratopological groups is feebly compact, and that a paratopological group G is feebly compact provided it has a feebly compact normal subgroup H such that a quotient group G/H is feebly compact. For our research we also study some general constructions of paratopological groups. We extend the well-known construction of Ra˘ıkov completion of a T0 topological group to the class of paratopological groups. We investigate cone topologies of paratopological groups which provide a general tool for constructing pathological examples, especially examples of compact-like paratopological groups with discontinuous inversion. We find a simple interplay between the algebraic properties of a basic cone subsemigroup S of a group G and compact-like properties of two basic semigroup topologies generated by S on the group G. -
Antidiamond Principles and Topological Applications 1
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 11, November 2009, Pages 5695–5719 S 0002-9947(09)04705-9 Article electronically published on June 24, 2009 ANTIDIAMOND PRINCIPLES AND TOPOLOGICAL APPLICATIONS TODD EISWORTH AND PETER NYIKOS Abstract. We investigate some combinatorial statements that are strong enough to imply that ♦ fails (hence the name antidiamonds); yet most of them are also compatible with CH. We prove that these axioms have many consequences in set-theoretic topology, including the consistency, modulo large cardinals, of a Yes answer to a problem on linearly Lindel¨of spaces posed by Arhangel’ski˘ı and Buzyakova (1998). 1. Introduction Researchers of set theory and allied branches of mathematics have long been fa- miliar with the contrast between the effects of the Continuum Hypothesis (CH) and those of MA+¬CH (Martin’s Axiom together with the negation of CH). Also long well-known is the way the contrast is greatly magnified when CH is strengthened to the combinatorial axiom ♦ while MA+¬CH is strengthened to the Proper Forc- ing Axiom (PFA). Until recently, however, comparatively little was known about just how much stronger ♦ really is than CH alone. Although a great variety of mathematical structures have been defined with the help of ♦, in most cases it was simply not known until recently whether or not CH alone is adequate to produce structures with the same basic properties. For many years, Shelah seemed to have a near-monopoly on models of CH+¬♦, as exemplified by the D-completeness constructions in [34]. The machinery of D- completeness is very difficult to apply directly, while other CH+¬♦ constructions prior to 1996 were of very limited applicability. -
The Discrete Countable Chain Condition
THE DISCRETE COUNTABLE CHAIN CONDITION MARGARET REAMES WI SCAMP, I. Introduction. In this paper, we investigate a topological prop- erty, the discrete countable chain condition (DCCC), which is a generalization of separability, of the Lindelof property, and of count- able compactness. These relationships are studied in §2 and this property is shown to be equivalent to an apparently weaker condition (Theorem 2.6). A topological space R is said to satisfy the countable chain condition (CCC) if every collection of disjoint open sets is countable. A topological space R is said to satisfy the discrete countable chain condition (DCCC) if every discrete collection of open sets is count- able. It is well known that a separable space satisfies the countable chain condition, and it is easy to find an example to show that the converse fails [6, p. 60]. That CCC implies DCCC is obvious from the defini- tions. An example of E. Michael [7] gives us a topological space that satisfies the DCCC but not the CCC. This space is Lindelof, hence satisfies the DCCC (Theorem 2.1), but has an uncountable subset with the discrete topology, hence does not satisfy the CCC. II. Relations between the DCCC and other topological properties. There are examples of Lindelof spaces wdiich do not satisfy the CCC and spaces satisfying the CCC which are not Lindelof, however, Theorem 2.1 gives us a relation between a Lindelof space and a space satisfying the DCCC. For convenience in stating the first few theorems, we label three conditions as follows: (i) R is Lindelof. -
The Equivalence of Two Definitions of Sequential Pseudocompactness
Appl. Gen. Topol. 17, no. 1(2016), 1-5 @ doi:10.4995/agt.2016.4616 c AGT, UPV, 2016 The equivalence of two definitions of sequential pseudocompactness Paolo Lipparini a Dipartimento di Matematica, Viale della Ricerca Scient`ıfica, II Universit`aGelmina di Roma (Tor Vergata), I-00133 Rome, Italy ([email protected]) Abstract We show that two possible definitions of sequential pseudocompactness are equivalent, and point out some consequences. 2010 MSC: Primary 54D20; Secondary 54B10. Keywords: sequentially pseudocompact; feebly compact; topological space. 1. The equivalence According to Artico, Marconi, Pelant, Rotter and Tkachenko [1, Definition 1.8], a Tychonoff topological space X is sequentially pseudocompact if the fo- llowing condition holds. (1) For any family (On)n∈ω of pairwise disjoint nonempty open sets of X, there are an infinite set J ⊆ ω and a point x ∈ X such that every neighborhood of x intersects all but finitely many elements of (On)n∈J . Notice that in [1] X is assumed to be a Tychonoff space, but the above definition makes sense for an arbitrary topological space. According to Dow, Porter, Stephenson, and Woods [2, Definition 1.4], a topological space is sequentially feebly compact if the following condition holds. (2) For any sequence (On)n∈ω of nonempty open subsets of X, there are an infinite set J ⊆ ω and a point x ∈ X such that every neighborhood of x intersects all but finitely many elements of (On)n∈J . (the difference is that in Condition (1) the On’s are assumed to be pairwise disjoint, while they are arbitrary in Condition (2)) Received 10 October 2013 – Accepted 11 September 2015 P.