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.