Sequence-Covering Maps of Metric Spaces

Total Page:16

File Type:pdf, Size:1020Kb

Sequence-Covering Maps of Metric Spaces Topology and its Applications 109 (2001) 301–314 Sequence-covering maps of metric spaces Shou Lin a;∗, Pengfei Yan b;1 a Department of Mathematics, Ningde Teacher’s College, Ningde, Fujian 352100, People’s Republic of China b Department of Mathematics, Anhui University, Hefei 230039, People’s Republic of China Received 26 March 1999; received in revised form 23 July 1999; accepted 12 August 1999 Abstract Let f : X ! Y be a map. f is a sequence-covering map if whenever fyng is a convergent sequence −1 in Y , there is a convergent sequence fxng in X with each xn 2 f .yn/. f is a 1-sequence-covering −1 map if for each y 2 Y ,thereisx 2 f .y/ such that whenever fyng is a sequence converging to y in −1 Y there is a sequence fxng converging to x in X with each xn 2 f .yn/. In this paper we investigate the structure of sequence-covering images of metric spaces, the main results are that (1) every sequence-covering, quotient and s-image of a locally separable metric space is a local @0-space; (2) every sequence-covering and compact map of a metric space is a 1-sequence-covering map. 2001 Elsevier Science B.V. All rights reserved. Keywords: Sequence-covering maps; 1-sequence-covering maps; Quotient maps; cs-networks; Weak bases; Point-countable covers; Sequential neighborhoods; Sequential spaces AMS classification: Primary 54E99; 54C10, Secondary 54D55 1. Introduction “Mappings and spaces” is one of the questions in general topology [1]. Spaces with certain k-networks play an important role in the theory of generalized metric spaces [7,8]. In the past the relations among spaces with certain k-networks were established by means of maps [1,11], in which quotient maps, closed maps, open maps and compact-covering maps were powerful tool. In recent years, sequence-covering maps introduced by Siwiec in [25] cause attention once again [12,14,16,17,19,30,31,33,36]. Partly, that is because * Corresponding author. Current address: Department of Mathematics, Fujian Normal University, Fuzhou 350007, People’s Republic of China. The project supported by NSFC (No. 19501023) and NSF of Fujian Province in China (No. A97025). Fax: +86-593-2954127, Tel.: +86-593-2955127. 1 Current address: Department of Mathematics, Shandong University, Tinan 250100, People’s Republic of China. 0166-8641/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII:S0166-8641(99)00163-7 302 S. Lin, P. Yan / Topology and its Applications 109 (2001) 301–314 sequence-covering maps are closely related to the question about compact-covering and s- images of metric spaces [22,23], certain quotient images of metric spaces [29,30], and they are a suitable map associated metric spaces with spaces having certain cs-networks [14, 27]. The present paper contributes to the problem of characterizing the certain quotient images or sequence-covering images of metric spaces, which is inspired by the following questions. Question 1.1 [33]. What is a nice characterization for a quotient and s-image of a locally separable metric space? Question 1.2 [32]. For a sequential space X with a point-regular cs-network, characterize X by means of a nice image of a metric space. Those questions are motivated by the following assertions: (1) A space is a quotient and s-image of a metric space if and only if it is a sequential space with a point-countable cs*-network [29]; (2) A space is an open and compact image of a metric space if and only if it has a point-regular base [1]. First, recall some basic definitions. All spaces are considered to be regular and T1,andall maps continuous and onto. Definition 1.3. Let f : X ! Y be a map. (1) f is an s-map if each f −1.y/ is separable. (2) f is a compact map if each f −1.y/ is compact. (3) f is a compact-covering map [21] if each compact subset of Y is the image of some compact subset of X. (4) f is a sequence-covering map [25] if each convergent sequence of Y is the image of some convergent sequence of X. (5) f is a sequentially quotient map [2] if for each convergent sequence L of Y ,there is a convergent sequence S of X such that f.S/is a subsequence of L. Definition 1.4. Let X be a space, and let P be a cover of X. (1) P is a network if whenever x 2 U with U open in X,thenx 2 P ⊂ U for some P 2 P. P ⊂ (2) is aSk-network [24] if whenever K U with K compact and U open in X,then K ⊂ P0 ⊂ U for some finite P0 ⊂ P. (3) P is a cs-network [10] if whenever fxng is a sequence converging to a point x 2 U with U open in X,thenfxg[fxn: n > m}⊂P ⊂ U for some m 2 N and some P 2 P. (4) P is a cs*-network [6] if whenever fxng is a sequence converging to a point x 2 U f g[f 2 N}⊂ ⊂ f g with U open in X,then x xni : i P U for some subsequence xni of fxng and some P 2 P. S. Lin, P. Yan / Topology and its Applications 109 (2001) 301–314 303 A space is a cosmic space [21] if it has a countable network. A space is an @0-space [21] if it has a countable k-network, and it is equivalent to a space with a countable cs-network or a countable cs*-network. Definition 1.5 [5]. Let X be a space. (1) Let x 2 P ⊂ X. P is a sequential neighborhood of x in X if whenever fxng is a sequence converging to the point x,thenfxn: n > m}⊂P for some m 2 N. (2) Let P ⊂ X. P is a sequentially open subset in X if P is a sequential neighborhood of x in X for each x 2 P . (3) X is a sequential space if each sequentially open subset in X is open. We recall that a cover P is point-countable if fP 2 P: x 2 P g is countable for each x 2 X, P is star-countable if fQ 2 P: Q \ P 6D ;g is countable for each P 2 P, P is locally countable if for each x 2 X, there is an open neighborhood V of x in X such that fP 2 P: P \ V 6D ;g is countable. A space X is metalindelöf if each open cover of X has a point-countable open refinement. 2. Sequential separability Definition 2.1 [28,37]. A space X is sequentially separable if X has a countable subset D such that for each x 2 X, there is a sequence fxng in D with xn ! x. D is called a sequentially dense subset of X. Liu and Tanaka [18] showed that every cosmic space with a point-countable cs-network is an @0-space, in which key step is that every cosmic space is sequentially separable. Michael [21] proved that a space X is a cosmic space if and only if it is an image of a separable metric space. We shall show that every sequentially separable space has a similar result. Recall some basic definitions. A space X is Fréchet [5], if whenever x 2 cl.A/ ⊂ X, there is a sequence in A converging to the point x. A space X is developable [4] if X has a development, i.e., there is a sequence fUng of open covers of X such that fst.x; Un/: n 2 Ng is a local base of x for each x 2 X. It is clear that developable spaces ) first countable spaces ) Fréchet spaces ) sequential spaces. Lemma 2.2. Sequential separability is preserved by a map. Lemma 2.3. Every separable and Fréchet space is sequentially separable. Lemma 2.4. Let X be a sequentially separable space. (1) If X has a point-countable cs*-network, X is a cosmic space. (2) If X has a point-countable k-network, X is a cosmic space. (3) If X has a point-countable cs-network, X is an @0-space. (4) If X has a star-countable k-network, X is an @0-space. 304 S. Lin, P. Yan / Topology and its Applications 109 (2001) 301–314 Proof. Let X be a sequentially separable space with a countable and sequentially dense subset D.IfX has a point-countable cs*-network P, put 0 P D P 2 P: P \ D 6D ; : 0 Then P is countable. For each x 2 U with U open in X, there is a sequence fxng in D with ! P f g[f 2 N}⊂ ⊂ xn x.Since is a cs*-network for X, x xni : i P U for some subsequence f g f g 2 P 2 P0 P0 xni of xn and some P ,andP . Thus is a countable network of X,andX is a cosmic space. If X has a point-countable k-network P, put 0 P D cl.P /: P 2 P;P\ D 6D ; : Then P0 is countable. For each x 2 U with U open in X, there are an open set V of X and a sequence fxng in D such that xn ! x 2 V ⊂ cl.V / ⊂ U.SinceP is a k-network f 2 N}⊂ ⊂ f g f g 2 P for X, xni : i P V for some subsequence xni of xn and some P ,and cl.P / 2 P0 and x 2 cl.P / ⊂ U.
Recommended publications
  • Spaces Determined by Point-Countable Covers
    Pacific Journal of Mathematics SPACES DETERMINED BY POINT-COUNTABLE COVERS GARY FRED GRUENHAGE,ERNEST A. MICHAEL AND YOSHIO TANAKA Vol. 113, No. 2 April 1984 PACIFIC JOURNAL OF MATHEMATICS Vol. 113, No. 2,1984 SPACES DETERMINED BY POINT-COUNTABLE COVERS G. GRUENHAGE, E. MICHAEL AND Y. TANAKA Recall that a collection 9 of subsets of X is point-countable if every x G X is in at most countably many ?Gf. Such collections have been studied from several points of view. First, in characterizing various kinds of j-images of metric spaces, second, to construct conditions which imply that compact spaces and some of their generalizations are metrizable, and finally, in the context of meta-Lindelόf spaces. This paper will make some contributions to all of these areas. 1. Introduction. A space1 X is determined by a cover £P, or 9 determines X,2 if U C X is open (closed) in X if and only if U Π P is relatively open (relatively closed) in P for every P E <•?. (For example, X is determined by <$ if <$ is locally finite or if {P°: P E <3>} covers X.) We will explore this notion in the context of point-countable covers. Within the class of λ -spaces, spaces determined by point-countable covers turn out to be related to spaces having a point-countable Λ -net- work. Recall that a cover ζP of X is a k-network for X if, whenever K C U with K compact and U open in X9 then K C U φ C U for some finite ^C^.
    [Show full text]
  • I-Sequential Topological Spaces∗
    Applied Mathematics E-Notes, 14(2014), 236-241 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ -Sequential Topological Spaces I Sudip Kumar Pal y Received 10 June 2014 Abstract In this paper a new notion of topological spaces namely, I-sequential topo- logical spaces is introduced and investigated. This new space is a strictly weaker notion than the first countable space. Also I-sequential topological space is a quotient of a metric space. 1 Introduction The idea of convergence of real sequence have been extended to statistical convergence by [2, 14, 15] as follows: If N denotes the set of natural numbers and K N then Kn denotes the set k K : k n and Kn stands for the cardinality of the set Kn. The natural densityf of the2 subset≤ Kg is definedj j by Kn d(K) = lim j j, n n !1 provided the limit exists. A sequence xn n N of points in a metric space (X, ) is said to be statistically f g 2 convergent to l if for arbitrary " > 0, the set K(") = k N : (xk, l) " has natural density zero. A lot of investigation has been donef on2 this convergence≥ andg its topological consequences after initial works by [5, 13]. It is easy to check that the family Id = A N : d(A) = 0 forms a non-trivial f g admissible ideal of N (recall that I 2N is called an ideal if (i) A, B I implies A B I and (ii) A I,B A implies B I.
    [Show full text]
  • On Compact-Covering and Sequence-Covering Images of Metric Spaces
    MATEMATIQKI VESNIK originalni nauqni rad 64, 2 (2012), 97–107 research paper June 2012 ON COMPACT-COVERING AND SEQUENCE-COVERING IMAGES OF METRIC SPACES Jing Zhang Abstract. In this paper we study the characterizations of compact-covering and 1-sequence- covering (resp. 2-sequence-covering) images of metric spaces and give a positive answer to the following question: How to characterize first countable spaces whose each compact subset is metrizable by certain images of metric spaces? 1. Introduction To find internal characterizations of certain images of metric spaces is one of the central problems in General Topology. In 1973, E. Michael and K. Nagami [16] obtained a characterization of compact-covering and open images of metric spaces. It is well known that the compact-covering and open images of metric spaces are the first countable spaces whose each compact subset is metrizable. However, its inverse does not hold [16]. For the first countable spaces whose each compact subset is metrizable, how to characterize them by certain images of metric spaces? The sequence-covering maps play an important role on mapping theory about metric spaces [6, 9]. In this paper, we give the characterization of a compact- covering and 1-sequence-covering (resp. 2-sequence-covering) image of a metric space, and positively answer the question posed by S. Lin in [11, Question 2.6.5]. All spaces considered here are T2 and all maps are continuous and onto. The letter N is the set of all positive natural numbers. Readers may refer to [10] for unstated definition and terminology. 2.
    [Show full text]
  • Arxiv:1702.07867V1 [Math.FA]
    TOPOLOGICAL PROPERTIES OF STRICT (LF )-SPACES AND STRONG DUALS OF MONTEL STRICT (LF )-SPACES SAAK GABRIYELYAN Abstract. Following [2], a Tychonoff space X is Ascoli if every compact subset of Ck(X) is equicontinuous. By the classical Ascoli theorem every k- space is Ascoli. We show that a strict (LF )-space E is Ascoli iff E is a Fr´echet ′ space or E = ϕ. We prove that the strong dual Eβ of a Montel strict (LF )- space E is an Ascoli space iff one of the following assertions holds: (i) E is a ′ Fr´echet–Montel space, so Eβ is a sequential non-Fr´echet–Urysohn space, or (ii) ′ ω D E = ϕ, so Eβ = R . Consequently, the space (Ω) of test functions and the space of distributions D′(Ω) are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively. 1. Introduction. The class of strict (LF )-spaces was intensively studied in the classic paper of Dieudonn´eand Schwartz [3]. It turns out that many of strict (LF )-spaces, in particular a lot of linear spaces considered in Schwartz’s theory of distributions [18], are not metrizable. Even the simplest ℵ0-dimensional strict (LF )-space ϕ, n the inductive limit of the sequence {R }n∈ω, is not metrizable. Nyikos [16] showed that ϕ is a sequential non-Fr´echet–Urysohn space (all relevant definitions are given in the next section). On the other hand, Shirai [20] proved the space D(Ω) of test functions over an open subset Ω of Rn, which is one of the most famous example of strict (LF )-spaces, is not sequential.
    [Show full text]
  • Analytic Spaces and Their Tukey Types by Francisco Javier Guevara Parra a Thesis Submitted in Conformity with the Requirements F
    Analytic spaces and their Tukey types by Francisco Javier Guevara Parra A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright ⃝c 2019 by Francisco Javier Guevara Parra Abstract Analytic spaces and their Tukey types Francisco Javier Guevara Parra Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2019 In this Thesis we study topologies on countable sets from the perspective of Tukey reductions of their neighbourhood filters. It turns out that is closely related to the already established theory of definable (and in particular analytic) topologies on countable sets. The connection is in fact natural as the neighbourhood filters of points in such spaces are typical examples of directed sets for which Tukey theory was introduced some eighty years ago. What is interesting here is that the abstract Tukey reduction of a neighbourhood filter Fx of a point to standard directed sets like or `1 imposes that Fx must be analytic. We develop a theory that examines the Tukey types of analytic topologies and compare it by the theory of sequential convergence in arbitrary countable topological spaces either using forcing extensions or axioms such as, for example, the Open Graph Axiom. It turns out that in certain classes of countable analytic groups we can classify all possible Tukey types of the corresponding neighbourhood filters of identities. For example we show that if G is a countable analytic k-group then 1 = f0g; and F G: are the only possible Tukey types of the neighbourhood filter e This will give us also new metrization criteria for such groups.
    [Show full text]
  • Hausdorff Separability of the Boundaries for Spacetimes And
    Hausdorff separability of the boundaries for spacetimes and sequential spaces. J.L. Flores∗, J. Herreraz, M. S´anchezy ∗Departamento de Algebra,´ Geometr´ıay Topolog´ıa, Facultad de Ciencias, Universidad de M´alaga, Campus Teatinos, 29071 M´alaga,Spain zInstituto de Matem´atica e Estat´ıstica, Universidade de S~aoPaulo, Rua do Mat~ao,1010, Cidade Universitaria S~aoPaulo, Brazil yDepartamento de Geometr´ıay Topolog´ıa, Facultad de Ciencias, Universidad de Granada, Avenida Fuentenueva s/n, 18071 Granada, Spain January 6, 2015 There are several ideal boundaries and completions in General Relativity sharing the topological property of being sequential, i.e., determined by the convergence of its sequences and, so, by some limit operator L. As emphasized in a classical article by Geroch, Liang and Wald, some of them have the property, commonly regarded as a drawback, that there are points of the spacetime M non T1-separated from points of the boundary @M. Here we show that this problem can be solved from a general topological viewpoint. In particular, there is a canonical minimum refinement of the topology in the comple- tion M which T2-separates the spacetime M and its boundary @M |no matter the type of completion one chooses. Moreover, we analyze the case of sequential spaces and show how the refined T2-separating topology can be constructed from a modifi- cation L∗ of the original limit operator L. Finally, we particularize this procedure to the case of the causal boundary and show how the separability of M and @M can be introduced as an abstract axiom in its definition.
    [Show full text]
  • Sequential Properties of Function Spaces with the Compact-Open Topology
    SEQUENTIAL PROPERTIES OF FUNCTION SPACES WITH THE COMPACT-OPEN TOPOLOGY GARY GRUENHAGE, BOAZ TSABAN, AND LYUBOMYR ZDOMSKYY Abstract. Let M be the countably in¯nite metric fan. We show that Ck(M; 2) is sequential and contains a closed copy of Arens space S2. It follows that if X is metrizable but not locally compact, then Ck(X) contains a closed copy of S2, and hence does not have the property AP. We also show that, for any zero-dimensional Polish space X, Ck(X; 2) is sequential if and only if X is either locally compact or the derived set X0 is compact. In the case that X is a non-locally compact Polish space whose derived set is compact, we show that all spaces Ck(X; 2) are homeomorphic, having the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n + 1)st. 1. Introduction Let Ck(X) be the space of continuous real-valued functions on X with the compact-open topology. Ck(X) for metrizable X is typically not a k-space, in particular not sequential. Indeed, by a theorem of R. Pol [8], for X paracompact ¯rst countable (in particular, metrizable), Ck(X) is a k-space if and only if X is locally compact, in which case X is a topological sum of locally compact σ-compact spaces and Ck(X) is a product of completely metrizable spaces. A similar result holds for Ck(X; [0; 1]): it is a k-space if and only if X is the topological sum of a discrete space and a locally compact σ-compact space, in which case Ck(X) is the product of a compact space and a completely metrizable space.
    [Show full text]
  • Sequential Conditions and Free Products of Topological Groups Sidney A
    PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 2, June 1988 SEQUENTIAL CONDITIONS AND FREE PRODUCTS OF TOPOLOGICAL GROUPS SIDNEY A. MORRIS AND H. B. THOMPSON (Communicated by Dennis Burke) ABSTRACT. If A and B are nontrivial topological groups, not both discrete, such that their free product A 11 B is a sequential space, then it is sequential of order oji. 1. Introduction. In [15], Ordman and Smith-Thomas prove that if the free topological group on a nondiscrete space is sequential then it is sequential of order uji. In particular, this implies that free topological groups are not metrizable or even Fréchet spaces. Our main result is the analogue of this for free products of topological groups. More precisely we prove that if A and B are nontrivial topological groups not both discrete and their free product A II B is sequential, then it is sequential of order ui. This result is then extended to some amalgamated free products. Our theorem includes, as a special case, the result of [10]. En route we extend the Ordman and Smith-Thomas result to a number of other topologies on a free group including the Graev topology which is the finest locally invariant group topology. (See [12].) It should be mentioned, also, that O.dman and Smith- Thomas show that the condition of AII B being sequential is satisfied whenever A and B are sequential fc^-spaces. 2. Preliminaries. The following definitions and examples are based on Franklin [3, 4] and Engelking [2]. DEFINITIONS. A subset U of a topological space X is said to be sequentially open if each sequence converging to a point in U is eventually in U.
    [Show full text]
  • Fasciculi Mathemat
    FASCICULIMATHEMATICI Nr 55 2015 DOI:10.1515/fascmath-2015-0024 Luong Quoc Tuyen MAPPING THEOREMS ON SPACES WITH sn-NETWORK g-FUNCTIONS Abstract. Let ∆ be the sets of all topological spaces satisfying the following conditions. (1) Each compact subset of X is metrizable; (2) There exists an sn-network g-function g on X such that if xn ! x and yn 2 g(n; xn) for all n 2 N, then x is a cluster point of fyng. In this paper, we prove that if X 2 ∆, then each sequentially- quotient boundary-compact map on X is pseudo-sequence-cove- ring; if X 2 ∆ and X has a point-countable sn-network, then each sequence-covering boundary-compact map on X is 1-sequence-covering. As the applications, we give that each sequentially-quotient boundary-compact map on g-metrizable spaces is pseudo-sequence-covering, and each sequence-covering boundary-compact on g-metrizable spaces is 1-sequence-covering. Key words: sn-networks, sn-network g-functions, g-metrizable spaces, boundary-compact maps, sequentially-quotient maps, pseudo-sequence-covering maps, sequence-covering maps, 1-se- quence-covering maps. AMS Mathematics Subject Classification: 54C10, 54E40, 54E99. 1. Introduction and preliminaries A study of images of topological spaces under certain sequence-covering maps is an important question in general topology. In 2001, S. Lin and P. Yan proved that each sequence-covering and compact map on metric spaces is 1-sequence-covering ([15]). Furthermore, S. Lin proved that each sequentially-quotient compact maps on metric spaces is pseudo-sequence- covering, and there exists a sequentially-quotient π-map on metric spaces is not pseudo-sequence-covering ([14]).
    [Show full text]
  • Closed Maps and the Character of Spaces
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 90. Number 2. February 1984 CLOSED MAPS AND THE CHARACTER OF SPACES YOSHIO TANAKA Abstract. We give some necessary and sufficient conditions for the character of spaces to be preserved under closed maps. Introduction. Recall the following result due to K. Morita and S. Hanai [11] or A. H. Stone [14]: Let Y be the image of a metric space X under a closed map/. Then Y is first countable (or metrizable) if and only if every boundary df~x(y) of the point-inverse/"'^) is compact. E. Michael [8] showed that every 3/~x(y) is compact if Xis paracompact, and Fis locally compact or first countable. For the Lindelöfness of 3/"'(y) with X metric, see [15]. For a space X and x G X, let x(x, X) be the smallest cardinal number of the form | %(x) |, where ^(x) is a nbd base at x in X. The character x( X) of X is defined as the supremum of all numbers \(x, X) for x G X. Let/be a closed map from X onto Y. First, we show that the character of Y has an influence on the boundaries 3/~'(y); indeed, they become Lindelöf or a-compact by the situation of Y. Second, in terms of these boundaries, we give some necessary and sufficient conditions for the character of X to be preserved under/. We assume all spaces are regular and all maps are continuous and onto. 1. a-compactness of the boundaries. We recall some definitions. A space X is strongly collectionwise Hausdorff if, whenever D — {xa; a G A} is a discrete closed subset of X, there is a discrete collection {Ua; a G A) of open subsets with Ua D D = {xa}.
    [Show full text]
  • A Study on Sequential Spaces in Topological and Ideal Topological Spaces
    A STUDY ON SEQUENTIAL SPACES IN TOPOLOGICAL AND IDEAL TOPOLOGICAL SPACES Synopsis submitted to Madurai Kamaraj University in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy in Mathematics by P. Vijaya Shanthi (Registration No. F 9832) under the Guidance of Dr. V. Renuka Devi Assistant Professor Centre for Research and Post Graduate Studies in Mathematics Ayya Nadar Janaki Ammal College (Autonomous, Affiliated to Madurai Kamaraj University, Re-accredited (3rd cycle) with `A' Grade (CGPA 3.67 out of 4) by NAAC, recognized as College of Excellence & Mentor College by UGC, STAR College by DBT and Ranked 51st at National Level in NIRF 2019) Sivakasi - 626 124 Tamil Nadu India March - 2020 SYNOPSIS The thesis entitled \A study on sequential spaces in topological and ideal topological spaces" consists of five chapters. In this thesis, we discuss about statistically convergence of sequences, se- quentially I-convergence structure and generalized convergence of a sequence in topological and ideal topological spaces. First chapter entitled \Preliminaries" consisting of basic definitions and some results which are used in this thesis. The idea of statistical convergence for real sequence was introduced by Fast [5], Steinhaus [13]. Cakalh [1] gave characterizations of statistical convergence for bounded sequences. Recently, Pehlivan and Mamedov [12] proved that all optimal paths have the same unique statistical cluster point which is also a statistical limit point. Some basic properties of statistical convergence were proved by Connor and Kline [3]. In general, statistical convergent sequences statisfy many of the properties of ordinary convergent sequences in metric spaces [8].
    [Show full text]
  • SPACES with Σ-LOCALLY FINITE LINDELÖF Sn-NETWORKS Luong
    PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 93 (107) (2013), 145–152 DOI: 10.2298/PIM1307145T SPACES WITH σ-LOCALLY FINITE LINDELÖF sn-NETWORKS Luong Quoc Tuyen Communicated by Miloš Kurilić Abstract. We prove that a space X has a σ-locally finite Lindelöf sn-network if and only if X is a compact-covering compact and mssc-image of a locally separable metric space, if and only if X is a sequentially-quotient π and mssc-image of a locally separable metric space, where “compact-covering” (or “sequentially-quotient”) can not be replaced by “sequence-covering”. As an application, we give a new characterization of spaces with locally countable weak bases. 1. Introduction In [17] Lin introduced the concept of mssc-maps to characterize spaces with certain σ-locally finite networks by mssc-images of metric spaces. After that, some characterizations for certain mssc-images of metric (or semi-metric) spaces are ob- tained by many authors ([11, 12, 14], for example). Recently, Dung gave some characterizations for certain mssc-images of locally separable metric spaces (see in [3]). We prove that a space X has a σ-locally finite Lindelöf sn-network if and only if X is a compact-covering compact and mssc-image of a locally separable metric space, if and only if X is a sequentially-quotient π and mssc-image of a locally separable metric space, where “compact-covering” (or “sequentially-quotient”) can not be replaced by “sequence-covering”. As an application, we give a new charac- terization of spaces with locally countable weak bases.
    [Show full text]