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Topology Proceedings Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT °c by Topology Proceedings. All rights reserved. TOPOLOGY PROCEEDINGS Volume 27, No. 1, 2003 Pages 351{360 ON COMPACT IMAGES OF LOCALLY SEPARABLE METRIC SPACES GE YING Abstract. In this paper, we give a characterization of seq- uentially-quotient compact images of locally separable metric spaces to prove that a space X is a sequentially-quotient com- pact image of a locally separable metric space if and only if X is a pseudo-sequence-covering compact image of a locally sep- arable metric space. As an application of the above result, we obtain that a space X is a quotient compact image of a locally separable metric space if and only if X is a pseudo-sequence- covering quotient compact image of a locally separable metric space, which answers a question posed by Y. Ikeda. 1. Introduction To determine what spaces are the images of \nice" spaces under \nice" mappings is one of the central questions of general topology (see, e.g., [1]). In the past, some noteworthy results on sequence- covering images of metric spaces have been obtained (see [4], [9], [10], [11], [12], [13], [16], [17]). Notice that pseudo-sequence-covering mapping and sequentially-quotient mapping are two important gen- eralizations of sequence-covering mapping. In his book, S. Lin [8] asked: If the domains are (locally separable) metric spaces, are 2000 Mathematics Subject Classification. Primary 54C10, 54D50, 54D55; Secondary 54E99. Key words and phrases. compact mapping, locally separable metric space, point-star network, sequentially-quotient (pseudo-sequence-covering, subsequence-covering) mapping. The author was supported in part by NSF of the Education Committee of Jiangsu Province in China #98KJB110005. 351 352 G. YING pseudo-sequence-covering compact mappings equivalent to sequen- tially-quotient compact mappings? This question is still open, which arouses our interest in the relations between pseudo-sequence- covering compact images and sequentially-quotient compact images for these metric domains. In [16], P. Yan proved that pseudo- sequence-covering compact images are equivalent to sequentially- quotient compact images for metric domains. Later, Lin and Yan [10] proved that pseudo-sequence-covering compact images are equiv- alent to sequentially-quotient compact images for separable metric domains. Thus, it is natural to raise the following question, which is helpful in solving [8, Question 3.4.8]. Question 1.1. Are pseudo-sequence-covering compact images equivalent to sequentially-quotient compact images for locally sep- arable metric domains? Finding the internal characterizations of certain images of metric spaces is also of considerable interest in general topology. Recently, many topologists were engaged in research of internal characteri- zations of compact images of metric spaces and separable metric spaces, and some noteworthy results have been obtained (see [2], [4], [6], [10], [11], [13], [14], [15], [16]). This leads us to be interested in compact images of locally separable metric spaces ([8]), that is, we are interested in the following question. Question 1.2. How are compact images of locally separable metric spaces characterized? Related to the above question, Lin, C. Liu, and M. Dai [9] gave an answer on quotient compact images of locally separable metric spaces, and so far there are no other results. In this paper, we give an internal characterization of sequentially- quotient compact images of locally separable metric spaces, and make use of the characterization to prove that a space X is a sequentially-quotient compact image of a locally separable metric space if and only if X is a pseudo-sequence-covering compact im- age of a locally separable metric space. As an application of the above result, we obtain that a space X is a quotient compact image of a locally separable metric space if and only if X is a pseudo- sequence-covering quotient compact image of a locally separable metric space,which answers a question posed by Y. Ikeda in [5]. COMPACT IMAGES 353 Throughout this paper, all spaces are T2 and all mappings are continuous and onto. N and ! denote the set of all natural numbers and first infinite ordinal, respectively. Let A be a subset of a space X, x X, be a family of subsets of X, and f be a mapping. 2 U We write ( )x = U : x U , st(x; ) = U : x U , st(A; ) =U U f 2: UU 2A =g φ , U A [f= U2 U A :2U g U [f 2 U \ 6 g U\ f \ 2 , f( ) = f(U): U . The sequence xn : n N , the Ug U f 2 Ug f 2 g sequence Pn : n N of subsets, and the sequence n : n f 2 g fP 2 N of families of subsets are abbreviated xn , Pn , and n , respectively.g Notice that definitions of somef mappingsg f g are differentfP g in different references. Definitions of mappings in this paper are quoted from [10], [8], [13], [16]. For terms which are not defined here, please refer to [3]. Definition 1.3 ([8]). Let X be a space, x P X. P is said to be 2 ⊂ a sequential neighborhood of x, if every sequence xn converging f g to x is eventually in P ; i.e., there is k N such that xn P for n > k. 2 2 Remark 1.4. (1) P is a sequential neighborhood of x iff x P 2 and every sequence xn converging to x is cofinally in P ; i.e., for f g each k N, there is n > k such that xn P . (2) The2 intersection of finitely many sequential2 neighborhoods of x is a sequential neighborhood of x. Definition 1.5 ([11]). Let n be a sequence of covers of a space X. fP g (1) n is a point-star network of X, if st(x; n) is a network at x infPX gfor each x X; f P g 2 (2) n is a refinement of X, if n+1 refines n for each n N; fP g P P 2 (3) n is point-finite, if n is point-finite for each n N. fP g P 2 We suppose every convergent sequence in the following defini- tions contains its limit point. Definition 1.6 ([2], [9], [16]). Let f : X Y be a mapping. 1 −! (1) f is compact, if f − (y) is compact in X for each y Y ; (2) f is sequence-covering (pseudo-sequence-covering)1,2 if for ev- ery convergent sequence S in Y , there is a convergent sequence L in X (compact subset K in X) such that f(L) = S (f(K) = S). 1\pseudo-sequence-covering" was called \sequence-covering" by E. Michael. 354 G. YING (3) f is sequentially-quotient (subsequence-covering), if for every convergent sequence S in Y , there is a convergent sequence L in X (compact subset K in X) such that f(L)(f(K)) is an infinite subsequence of S. Remark 1.7. The following implications are obvious by Definition 1.6 and [8] and cannot be reversed. (1) sequence-covering mapping= pseudo-sequence-covering map- ping and sequentially-quotient mapping;) (2) pseudo-sequence-covering mapping= subsequence-covering mapping; ) (3) sequentially-quotient mapping= subsequence-covering map- ping. ) 2. Main Results Lin [7, Proposition 2.1.7] proved that pseudo-sequence-covering mappings on spaces in which points are Gδ0 s are sequentially-quo- tient mappings. In fact, pseudo-sequence-covering mappings in this result can be relaxed to subsequence-covering mappings. Proposition 2.1. Let f : X Y be a subsequence-covering map- −! ping, where points in X are Gδ. Then, f is sequentially-quotient. Proof: Let S be a sequence converging to y in Y . f is subse- quence-covering, so there is a compact subset K in X such that f(K) = S0 is a subsequence of S. Put S0 = y yn : n N , 1 f g [ f 2 g where yn converging y. Pick xn f (yn) K. Then, xn K. f g 2 − \ f g ⊂ Notice that K is a compact subspace in which points are Gδ0 s. Thus, K is first countable, hence sequentially compact, so there is 1 a subsequence xn of xn converging to x f (y). This proves f k g f g 2 − that f is sequentially-quotient. Theorem 2.2. For a space X, the following are equivalent. (1) X is a pseudo-sequence-covering compact image of a locally separable metric space; (2) X is a subsequence-covering compact image of a locally sep- arable metric space; (3) X is a sequentially-quotient compact image of a locally sep- arable metric space; COMPACT IMAGES 355 (4) X has a point-finite cover Xα : α Λ , where every Xα has f 2 g a sequence α,n of countable and point-finite refinements, and the followingfP conditionsg (a) and (b) are satisfied. (a) n is a point-star network of X, where n = α,n : α Λ ; fP g P [fP 2 g (b) for each x X, there is finite subset Λ0 of Λ such that st(x; (n; Λ )) is a2 sequential neighborhood of x for each n !, P 0 2 where (n; Λ ) = α,n : α Λ . P 0 [fP 2 0g Proof: (1) = (2) = (3) by Remark 1.7 and Proposition 2.1. We prove only) (3) = (4)) = (1). (3) = (4). Let f): M ) X be sequentially-quotient, and M ) −! be a locally separable metric space. By [3, 4.4.F], M = α ΛMα, ⊕ 2 where every Mα is separable.
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