DOCSLIB.ORG

Homeomorphism Groups of Commutator Width One

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Homeomorphism Groups of Commutator Width One PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 5, May 2013, Pages 1839–1847 S 0002-9939(2012)11595-3 Article electronically published on November 28, 2012 HOMEOMORPHISM GROUPS OF COMMUTATOR WIDTH ONE TAKASHI TSUBOI (Communicated by Alexander N. Dranishnikov) n Abstract. We show that any element of the identity component Homeo(S )0 of the group of homeomorphisms of the n-dimensional sphere Sn can be written as one commutator. We also show that any element of the group Homeo(μn) of homeomorphisms of the n-dimensional Menger compact space μn can be written as one commutator. 1. Introduction The algebraic properties of the group of homeomorphisms or diffeomorphisms are studied by many people. The identity components of the group of homeomorphisms or diffeomorphisms of compact manifolds are known to be perfect and moreover simple ([23], [1], [12], [10], [13], [16], [19], [11], [2]). Many of them, for example the group of homeomorphisms of the n-dimensional sphere, are known to be uniformly perfect ([1], [7], [20], [22]). A group is uniformly perfect if every element can be written as a product of a bounded number of commutators. The least number of such bounds is called the commutator width of the group. In this paper we show that the commutator width of the identity component n n Homeo(S )0 of the group of homeomorphisms of the n-dimensional sphere S is one. n Theorem 1.1. Any element of Homeo(S )0 can be written as one commutator. We also show that the commutator width of the group Homeo(μn)ofhomeo- morphisms of the n-dimensional Menger compact space μn is one. Theorem 1.2. Any element of Homeo(μn) can be written as one commutator. n Anderson showed that in the group Homeoc(R ) of homeomorphisms of the n- dimensional Euclidean space with compact support, any element can be written as n one commutator ([1], [15]). Since any element f of Homeo0(S ) can be written as a product f = gh such that g and h are the identity on some nonempty open sets, n the fact that the commutator width of Homeoc(R )isoneimpliesthatf can be written as a product of two commutators. Received by the editors September 12, 2011. 2010 Mathematics Subject Classification. Primary 54H15, 54H20, 57S05; Secondary 20F65, 37B05, 57N50. Key words and phrases. Homeomorphism group, uniformly perfect, commutator subgroup. The author is partially supported by Grant-in-Aid for Scientific Research (A) 20244003, (S) 24224002, Grant-in-Aid for Exploratory Research 21654009, 24654011, Japan Society for Pro- motion of Science, and by the Global COE Program at the Graduate School of Mathematical Sciences, University of Tokyo. c 2012 American Mathematical Society Reverts to public domain 28 years from publication 1839 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1840 TAKASHI TSUBOI It is worth recalling the construction by Anderson ([1]). For given f ∈ n Homeoc(R ), we find a bounded ball U such that the support supp(f) ⊂ U.Then n n we can find an element g ∈ Homeoc(R ) such that g (U)(n ∈ Z) are disjoint and ∞ lim diam(gn(U)) = 0. Put F = gnfg−n.ThenwehavegFg−1 = f −1F .Thus n→∞ n=0 f = FgF−1g−1. We understand the meaning that the commutator width is one as follows. In the n case of Homeoc(R ), we see that for any element f,thereexistsg such that g and fg are conjugate. That is, g is dynamically so strong that fg and g have the same dynamics, and hence they are conjugate. n n InthecaseofHomeo0(S )orHomeo(μ ), we have the candidate which has the strong dynamics. The candidate is the topologically hyperbolic homeomorphism. A topologically hyperbolic homeomorphism is a homeomorphism h with one source n n s+ and one sink s− such that lim h (x)=s− and lim h (x)=s+ for x ∈ n→+∞ n→−∞ {s−,s+}. It seems true that the orientation-preserving topologically hyperbolic homeomorphisms are all conjugate, but we are not able to show it at the present. The topologically hyperbolic homeomorphisms we construct later are conjugate because they are constructed with nice fundamental domains outside the fixed points. Hence what we do in this paper for a given homeomorphism f is to construct a topologically hyperbolic homeomorphism g whichissostrongthatfg is topologi- cally hyperbolic. We will prove Theorem 1.1 in Section 2 and Theorem 1.2 in Section 3. Added in proof : Shigenori Matsumoto told the author that the fact that the com- 1 mutator width of Homeo(S )0 is one follows from a result in Transverse foliations of Seifert bundles and self-homeomorphism of the circle, Comment. Math. Helvetici 56 (1981) 638–660, by D. Eisenbud, U. Hirsch and W. Neumann. 2. The group of homeomorphisms of the n-dimensional sphere For the proof of Theorem 1.1, we need the following deep theorems. Theorem 2.1 (Generalized Schoenflies Theorem, [4], [5]). Let Σ be a locally flat (n − 1)-dimensional sphere in the n-dimensional sphere Sn. Then the closures of the complementary domains of Σ are homeomorphic to the n-dimensional disk Dn. Here, an (n − 1)-dimensional submanifold Ln−1 in an n-dimensional manifold M n is locally flat if each point of Ln−1 has a neighborhood U in M n such that the pair (U, U ∩ Ln−1)ishomeomorphicto(Rn, Rn−1). Theorem 2.2 (Annulus conjecture, [14], [17]). Let Σ0 and Σ1 be disjoint locally flat (n − 1)-dimensional spheres in the n-dimensional sphere Sn. Then the closure of the region between them is homeomorphic to Sn−1 × [0, 1]. Let f be an orientation-preserving homeomorphism of the n-dimensional sphere Sn which is not the identity. Then we can find small n-dimensional closed disks n n n n n −1 n D0 and D1 in S such that ∂D0 , ∂D1 are locally flat and the four disks f (D0 ), n n n D0 , D1 , f(D1 )aredisjoint. n n\ n n n\ n By Theorem 2.1, D0 and S int(D1 ) are homeomorphic and D1 and S int(D0 ) are homeomorphic. Hence there exists an orientation-preserving homeomorphism n n n \ n n \ n n g of S such that g(D0 )=S int(D1 )andg(S int(D0 )) = D1 . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HOMEOMORPHISM GROUPS OF COMMUTATOR WIDTH ONE 1841 (gfg)(Σ) (fgf−1)(Σ) (g2)(Σ) (fg2)(Σ) f n −−→ n D1 f(D1 ) (gf−1)(Σ) (fgfg)(Σ) g(Σ) g (fg)(Σ) f −1(Σ) Σ (f −1g−1f −1)(Σ) (g−1f)(Σ) f −1 n −−→ n f (D0 ) D0 (f −1g−1)(Σ)(g−1)(Σ) (f −1g−1f)(Σ)(g−1f −1)(Σ) Figure 1. The actions of g and fg n − −1 Let Σ = ∂D0 . Thenwehavefourdisjoint(n 1)-dimensional spheres: f (Σ), −1 n n n n Σ, g(Σ), (fg)(Σ), which are the boundaries of f (D0 ),D0 ,D1 ,f(D1 ), respectively. Then we see that (gf−1)(Σ), (g2)(Σ), (gfg)(Σ)arecontained n −1 2 n −1 −1 in D1 ;(fgf )(Σ), (fg )(Σ), (fgfg)(Σ)arecontainedinf(D1 ); (g f )(Σ), −1 −1 n −1 −1 −1 −1 −1 (g )(Σ), (g fg)(Σ)arecontainedinD0 ;and(f g f )(Σ), (f g )(Σ), −1 −1 −1 n −1 n (f g f)(Σ)arecontainedinf (D0 ). See Figure 1, where f (D0 )islower n n n left, D0 is lower right, D1 is upper left and f(D1 ) is upper right, and where f −1 n n n n translates f (D0 )andD1 to D0 and f(D1 ), respectively. We require the homeomorphism g to satisfy the following conditions: (1) lim diam(gk(Σ)) = 0, k→±∞ (2) lim diam((fg)k(Σ)) = 0. k→±∞ n n n Let D denote the n-dimensional standard disk. Let ψ n : D −→ D and ψ n : D0 0 D1 n −→ n D D1 be homeomorphisms. Then by Theorem 2.1, we have homeomorphisms n n n n n ψ n\ n : D −→ S \ int(D ) extending ψ n |∂D and ψ n\ n : D −→ S int(D0 ) 0 D0 S int(D1 ) n \ n | n S int(D1 ) extending ψDn ∂D . We first define a homeomorphism g by 1 −1 n ψ n\ n ψ n on D , S int(D1 ) D0 0 g = −1 n n ψ n ψ n\ n on S \ int(D ) , D1 S int(D0 ) 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1842 TAKASHI TSUBOI andthenwemodifyg so that (1) and (2) are satisfied. For this purpose we notice the following fact. Lemma 2.3. For any compact set K in the interior int(Dn) of the standard disk k n n D and any positive real number ε, there is a homeomorphism ϕK,ε : D −→ D n which is the identity on ∂D such that diam(ϕK,ε(K)) ≤ ε. For a continuous map ψ between compact metric spaces ψ : X −→ Y ,let μψ denote the modulus of continuity of ψ. This means that distY (ψ(x),ψ(y)) ≤ μψ(distX (x, y)) for x, y ∈ X. The modification of g is done step by step. 2 −1 2 First we look at g (Σ)andtakeK1 = ψDn ((g )(Σ)) and ε1 such that μψ n (ε1) 1 D1 ≤ −2 2 2 ,whereμψ n is the modulus of continuity of ψDn .Thenwelookat(fg) (Σ) D1 1 −1 −2 and take K2 = ψDn ((gfg)(Σ)).
Load more

Recommended publications
  • 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.
    [Show full text]
  • Arxiv:1809.05508V1 [Math.GN] 14 Sep 2018 X 48,54H11
    A NON-DISCRETE SPACE X WITH Cp(X) MENGER AT INFINITY ANGELO BELLA AND RODRIGO HERNANDEZ-GUTI´ ERREZ´ Abstract. In a paper by Bella, Tokg¨os and Zdomskyy it is asked whether there exists a Tychonoff space X such that the remainder of Cp(X) in some compactification is Menger but not σ-compact. In this paper we prove that it is consistent that such space exists and in particular its existence follows from the existence of a Menger ultrafilter. 1. Introduction A space X is called Menger if for every sequence {Un ∶ n ∈ ω} of open covers of X one may choose finite sets Vn ⊂ Un for all n ∈ ω in such a way that ⋃{Vn ∶ n ∈ ω} covers X. Given a property P, a Tychonoff space X will be called P at infinity if βX ∖ X has P. Let X be a Tychonoff space. It is well-known that X is σ-compact at infinity if and only if X is Cech-complete.ˇ Also, Henriksen and Isbell in [7] proved that X is Lindel¨of at infinity if and only if X is of countable type. Moreover, the Menger property implies the Lindel¨of property and is implied by σ-compactness. So it was natural for the authors of [2] to study when X is Menger at infinity. Later, the authors of [4] study when a topological group is Menger, Hurewicz and Scheepers at infinity. The Hurewicz and Scheepers properties are other cov- ering properties that are stronger than the Menger property and weaker than σ- compactness (see the survey [12] by Boaz Tsaban).
    [Show full text]
  • 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].
    [Show full text]
  • STAR SELECTION PRINCIPLES: a SURVEY 1. Introduction There Are
    Khayyam J. Math. 1 (2015), no. 1, 82{106 STAR SELECTION PRINCIPLES: A SURVEY LJUBISAˇ D.R. KOCINACˇ Communicated by H.R. Ebrahimi Vishki Abstract. We review selected results obtained in the last fifteen years on star selection principles in topology, an important subfield of the field of selection principles theory. The results which we discuss concern also uniform structures and, in particular, topological groups and their generalizations. 1. Introduction There are many results in the literature which show that a number of topolog- ical properties can be characterized by using the method of stars. In particular it is the case with many covering properties of topological spaces. The method of stars has been used to study the problem of metrization of topological spaces, and for definitions of several important classical topological notions. More infor- mation on star covering properties can be found in [17], [45]. We use here such a method in investigation of selection principles for topological and uniform spaces. Although Selection Principles Theory is a field of mathematics having a rich his- tory going back to the papers by Borel, Menger, Hurewicz, Rothberger, Seirpi´nski in 1920{1930's, a systematic investigation in this area rapidly increased and attracted a big number of mathematicians in the last two-three decades after Scheeper's paper [54]. Nowadays, this theory has deep connections with many branches of mathematics such as Set theory and General topology, Game theory, Ramsey theory, Function spaces and hyperspaces, Cardinal invariants, Dimension theory, Uniform structures, Topological groups and relatives, Karamata theory. Researchers working in this area have organized four international mathemati- cal forums called \Workshop on Coverings, Selections and Games in Topology".
    [Show full text]
  • On the Homotopy Groups of Separable Metric Spaces ∗ F.H
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 158 (2011) 1607–1614 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol On the homotopy groups of separable metric spaces ∗ F.H. Ghane , Hadi Passandideh, Z. Hamed Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran article info abstract Article history: The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. Received 2 June 2010 First, we state a nerve theorem. It gives sufficient conditions under which there is a weak Received in revised form 12 May 2011 n-equivalence between the nerve of a good cover and its underlying space. Then we con- Accepted 19 May 2011 clude that for any (n − 1)-connected, locally (n − 1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, (X),is MSC: πn 55Q05 finitely presented. This result allows us to provide a new proof for a generalization of 55Q52 Shelah’s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 54E35 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topo- 57N15 logical space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic Keywords: 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite Nerve which introduce here.
    [Show full text]
  • 173–180 ALMOST MENGER and RELATED SPACES Darko Kocev 1
    MATEMATIQKI VESNIK UDK 515.122 61 (2009), 173–180 originalni nauqni rad research paper ALMOST MENGER AND RELATED SPACES Darko Kocev Abstract. In this paper we consider the notion of almost Menger property which is similar to the familiar property of Menger and prove that we can use regular open sets instead of open sets in the definition of almost Menger property. We give conditions for a space Xn to be almost Menger. In the similar way, we consider almost γ-sets and the almost star-Menger property. 1. Introduction and definitions In this paper we consider almost Menger and related spaces similar to the well-known property of Menger. For this we use the closures of open sets. The idea is not completely new. Tkachuk in [11] and Scheepers in [9] and implicitly in [10] considered a property similar to the classical notion of Rothberger [8] using the closures of open sets. In [5], Koˇcinacintroduced the notion of almost Menger property. In [6], Di Maio and Koˇcinacconsidered the almost Menger property in hyperspaces. We will show that the almost Menger property is different from the Menger property. In Section 2 we also show that we can replace open sets with regular open sets in the definition of almost Menger spaces. In Section 3 we have the characterization of almost Menger property in all finite powers. In Sections 4 and 5 we consider almost γ-sets and almost star-Menger spaces in the similar manner as almost Menger spaces in Section 2. We will assume that all topological spaces in this paper are Hausdorff.
    [Show full text]
  • Problems on Boundaries of Groups and Kleinian Groups
    PROBLEMS ON BOUNDARIES OF GROUPS AND KLEINIAN GROUPS MISHA KAPOVICH Most problems in this list were collected during the workshop \Boundaries in Geometric Group Theory", in AIM, 2005. 1. Background Ideal boundaries of hyperbolic spaces. Suppose that X is a hyperbolic metric space. Pick a base-point o 2 X. This de¯nes the Gromov product (x; y)o 2 R+ for points x; y 2 X. The ideal boundary @1X of X is the collection of equivalence classes [xi] of sequences (xi) in X where (xi) » (yi) if and only if lim (xi; yi)o = 1: i!1 The topology on @1X is de¯ned as follows. Let » 2 @1X. De¯ne r-neighborhood of » to be U(»; r) := f´ 2 @1X : 9(xi); (yi) with » = [xi]; ´ = [yi]; lim inf(xi; yj)o ¸ rg: i;j!1 Then the basis of topology at » consists of fU(»; r); r ¸ 0g. We will refer to the resulting ideal boundary @1X as the Gromov{boundary of X. One can check that the topology on @1X is independent of the choice of the base-point. Moreover, if f : X ! Y is a quasi-isometry then it induces a homeomorphism @1f : @1X ! @1Y . The Gromov product extends to a continuous function (»; ´)o : @1X £ @1X ! [0; 1]: a The geodesic boundary of X admits a family of visual metrics d1 de¯ned as follows. Pick a positive parameter a. Given points »; ´ 2 @1X consider various chains c = (»1; :::; »m) (where m varies) so that »1 = »; »m = ´. Given such a chain, de¯ne mX¡1 ¡a(»i;»i+1)o dc(»; ´) := e ; i=1 where e¡1 := 0.
    [Show full text]
  • Selected Results on Selection Principles
    Selected results on selection principles Ljubiˇsa D.R. Koˇcinac Abstract We review some selected recent results concerning selection princi- ples in topology and their relations with several topological construc- tions. 2000 Mathematics Subject Classification: 54-02, 54D20, 03E02, 05C55, 54A25, 54B20, 54C35, 91A44. Keywords: Selection principles, star selection principles, uniform selection principles, game theory, partition relations, relative properties, function spaces, hyperspaces. 1 Introduction The beginning of investigation of covering properties of topological spaces defined in terms of diagonalization and nowadays known as classical selection principles is going back to the papers [52], [35], [36], [63]. In this paper we shall briefly discuss these classical selection principles and their relations with other fields of mathematics, and after that we shall be concentrated on recent innovations in the field, preferably on results which are not included into two nice survey papers by M. Scheepers [73], [74]. In particular, in [74] some information regarding ”modern”, non-classical selection principles can arXiv:1201.1576v1 [math.GN] 7 Jan 2012 be found. No proofs are included in the paper. Two classical selection principles are defined in the following way. Let A and B be sets whose elements are families of subsets of an infinite set X. Then: Sfin(A, B) denotes the selection hypothesis: For each sequence (An : n ∈ N) of elements of A there is a sequence (Bn : n ∈ N) of finite sets such that for each n, Bn ⊂ An, and Sn∈N Bn is an element of B. 1 S1(A, B) denotes the selection principle: For each sequence (An : n ∈ N) of elements of A there is a sequence (bn : n ∈ N) such that for each n, bn ∈ An, and {bn : n ∈ N} is an element of B.
    [Show full text]
  • Arxiv:1806.10385V2 [Math.GN] 21 Sep 2018 Htfrmtial Pcshspoet Seuvln Ooeconside One to Equivalent Is Each Property [19]
    PRODUCTS OF MENGER SPACES IN THE MILLER MODEL LYUBOMYR ZDOMSKYY Abstract. We prove that in the Miller model the Menger property is preserved by finite products of metrizable spaces. This answers several open questions and gives another instance of the interplay between clas- sical forcing posets with fusion and combinatorial covering properties in topology. 1. Introduction A topological space X has the Menger property (or, alternatively, is a Menger space) if for every sequence hUn : n ∈ ωi of open covers of X <ω there exists a sequence hVn : n ∈ ωi such that Vn ∈ [Un] for all n and X = S{∪Vn : n ∈ ω}. This property was introduced by Hurewicz, and the current name (the Menger property) is used because Hurewicz proved in [15] that for metrizable spaces his property is equivalent to one considered by Menger in [19]. Each σ-compact space has obviously the Menger property, and the latter implies lindel¨ofness (that is, every open cover has a countable subcover). The Menger property is the weakest one among the so-called selection principles or combinatorial covering properties, see, e.g., [4, 17, 25, 26, 32] for detailed introductions to the topic. Menger spaces have recently found applications in such areas as forcing [12], Ramsey theory in algebra [34], combinatorics of discrete subspaces [2], and Tukey relations between hyperspaces of compacts [14]. In this paper we proceed our investigation of the interplay between posets with fusion and selection principles initiated in [24]. More precisely, we concentrate on the question whether the Menger property is preserved by finite products. For general topological spaces the answer negative: In ZFC arXiv:1806.10385v2 [math.GN] 21 Sep 2018 there are two normal spaces X,Y with a covering property much stronger than the Menger one such that X × Y does not have the Lindel¨of property, see [29, §3].
    [Show full text]
  • The Topological Degree of A-Proper Mapping in the Menger Pn-Space (I)
    BULL. AUSTRAL. MATH. SOC. 46S50, 47S50 VOL. 73 (2006) [161-168] THE TOPOLOGICAL DEGREE OF A-PROPER MAPPING IN THE MENGER PN-SPACE (I) HUANG XIAOQIN, WANG MIANSEN AND ZHU CHUANXI In this paper, we introduce the concept of A-proper topological degree in Menger PN-space and study some of its properties. Utilising its properties, we obtain a new fixed point theorem. 1. INTRODUCTION In 1940s, Menger advanced the concept of probabilistic metric space. In his theory, the distance between two points was represented by a distribution function. Obviously, compared with the structure of metric space, it further conformed to reality. Moreover, the ordinary metric space can be looked upon as its special cases. So, the study of probabilistic metric space has important practical significance. As everyone knows, the A-proper topological degree theory is a forceful tool in the research of operator theory in normed spaces. Then, how to establish and study the A-proper topological degree in probabilistic metric space? In this paper, we introduce the concept of A-proper topologi- cal degree in Menger PN- space and study some of its properties. Utilising its properties, we obtain a new fixed point theorem. For the sake of convenience, we recall some definitions and properties of PN-space. DEFINITION 1: (Chang [1].) A probabilistic normed space (shortly a PN-space) is an ordered pair (E,F), where E is a real linear space, F is a mapping of E into D (D is the set of all distribution functions. We shall denote the distribution function F(x) by Fx, Fx(i) denotes the value Fx for t € R) satisfying the following conditions: (PN-2) Fx(t) = H(t) for all t € R if and only if x = 0, where H(t)=0 when t s$ 0, and H(t)=l when t > 0; (PN-3) For all a ± 0, Fax(t) = Fx(t/\a\); (PN-4) For any x,y 6 E and tu t2 e R, if Fx(ti) = 1 and Fy{t2) - 1, then we have Fx+y{h+t2) = l.
    [Show full text]
  • 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.
    [Show full text]
  • Lindelöf Spaces Which Are Menger
    Lindel¨ofspaces which are Menger, Hurewicz, Alster, productive, or D Franklin D. Tall1 April 20, 2009 This paper is dedicated to Ken Kunen, who, in addition to his own ground-breaking research, has been insightfully guiding graduate students for 40 years. The dedication from my doctoral dissertation is still apt: he provides clear answers to murky questions. Abstract We discuss relationships in Lindel¨ofspaces among the properties \Menger", \Hurewicz", \Alster", \productive", and \D". This note is a continuation of [13]. The question of what additional assumptions ensure that the product of two Lindel¨ofspaces is Lindel¨ofis natural and well-studied. See e.g., [28], [30], [2], [3], [4], [5], [6], [32], [33]. D-spaces were introduced in [20]. Definition. A space X is D if for every neighbourhood assignment fVxgx2X , i.e. each Vx is an open set containing x, there is a closed discrete Y ⊆ X such that fVxgx2Y covers X. Y is called a kernel of the neighbourhood assignment. 1Research supported by grant A-7354 of the Natural Sciences and Engineering Research Council of Canada. (2000) Mathematics Subject Classification. Primary 54D45, 54D20, 54D99, 54A35, 03E35; Secondary 54G20, 54H05, 03E15, 03E17. Key words and phrases: Lindel¨of,productive, D, Alster, Hurewicz, Menger. 1 The question raised in [20] of whether every Lindel¨ofspace is a D-space has been surveyed in [22]. It has recently been the subject of much research. In [12], Aurichi established many connections among the D property, topological games, and selection properties. In this paper, we examine connections among selection principles, the D property, and preservation of Lindel¨ofnessunder products.
    [Show full text]