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

ON COMPACT IMAGES OF LOCALLY SEPARABLE METRIC SPACES

GE YING

Abstract. In this paper, we give a characterization of sequentially-quotient compact images of locally separable metric spaces 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 image 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.

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 sequentially-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 equivalent 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 separable 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 characterizations 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 image 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 ω 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. ∈ U We write ( )x = U : x U , st(x, ) = U : x U , st(A, ) =U U { ∈: UU ∈A =} φ , U A ∪{= U∈ U A :∈U } U ∪{ ∈ U ∩ 6 } U ∩ { ∩ ∈ , f( ) = f(U): U . The sequence xn : n N , the U} U { ∈ U} { ∈ } sequence Pn : n N of subsets, and the sequence n : n { ∈ } {P ∈ N of families of subsets are abbreviated xn , Pn , and n , respectively.} Notice that definitions of some{ mappings} { } are different{P } 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 ∈ ⊂ a sequential neighborhood of x, if every sequence xn converging { } to x is eventually in P ; i.e., there is k N such that xn P for n > k. ∈ ∈ Remark 1.4. (1) P is a sequential neighborhood of x iff x P ∈ and every sequence xn converging to x is cofinally in P ; i.e., for { } each k N, there is n > k such that xn P . (2) The∈ intersection of finitely many sequential∈ neighborhoods of x is a sequential neighborhood of x.

Definition 1.5 ([11]). Let n be a sequence of covers of a space X. {P } (1) n is a point-star network of X, if st(x, n) is a network at x in{PX }for each x X; { P } ∈ (2) n is a refinement of X, if n+1 refines n for each n N; {P } P P ∈ (3) n is point-finite, if n is point-finite for each n N. {P } P ∈ We suppose every convergent sequence in the following definitions 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,∈ 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) = 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 mapping and sequentially-quotient mapping;⇒ (2) pseudo-sequence-covering mapping= subsequence-covering mapping; ⇒ (3) sequentially-quotient mapping= subsequence-covering mapping. ⇒

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-quotient 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 subsequence-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 { } ∪ { ∈ } where yn converging y. Pick xn f (yn) K. Then, xn K. { } ∈ − ∩ { } ⊂ 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 { k } { } ∈ − 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 separable metric space; (3) X is a sequentially-quotient compact image of a locally separable metric space; COMPACT IMAGES 355

(4) X has a point-finite cover Xα : α Λ , where every Xα has { ∈ } a sequence α,n of countable and point-finite refinements, and the following{P conditions} (a) and (b) are satisfied. (a) n is a point-star network of X, where n = α,n : α Λ ; {P } P ∪{P ∈ } (b) for each x X, there is finite subset Λ0 of Λ such that st(x, (n, Λ )) is a∈ sequential neighborhood of x for each n ω, P 0 ∈ where (n, Λ ) = α,n : α Λ . P 0 ∪{P ∈ 0} 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, ⊕ ∈ where every Mα is separable. As M is metric, using [3, 5.4.E], there is a sequence n of locally finite open covers of M such that every {B } n+1 is a refinement of n, and for every compact subset K M B B ⊂ and any open set U K, there is n ω such that st(K, n) U. ⊃ ∈ B ⊂ For each n ω, there is a countable subfamily α,n of n, which ∈ B B covers Mα. We can assume α,n+1 refines α,n for each n N. For each α Λ, n ω, put B B ∈ ∈ ∈ Xα = f(Mα); α,n = f( α,n) ; n = α,n : α Λ . P B } P ∪{P ∈ } As f is compact, it is easy to check that Xα : α Λ is a { ∈ } point-finite cover of X, and α,n is a sequence of countable and {P } point-finite refinements of Xα for each α Λ. ∈ (a) n is a point-star network of X: {P } 1 1 1 Let x U and U be open in X. Then f − (x) f − (U). f − (x) ∈ ⊂ 1 is compact in M, so there is k ω such that st(f − (x), k) 1 ∈ 1 B ⊂ f − (U). It is easy to check x st(x, k) f(st(f − (x), k)) 1 ∈ P ⊂ B ⊂ ff (U) = U. So st(x, n) is a network at x. This proves that − { P } n is a point-star network of X. {P } For x X and n ω, put Λ = α Λ: x Xα , and ∈ ∈ 0 { ∈ ∈ } (n, Λ ) = α,n : α Λ . P 0 ∪{P ∈ 0} (b) st(x, (n, Λ0)) is a sequential neighborhood of x for each n ω: P ∈Let S be a sequence in X converging to x. f is sequentially- quotient, so there is a sequence L in M converging to t such that f(L) is a subsequence of S. There is α Λ such that t Mα . 0 ∈ 0 ∈ 0 M is a topological sum of Mα : α Λ , so Mα is open in M; { ∈ } 0 hence, L is eventually in Mα . α ,n is an open cover of Mα ; there 0 B 0 0 356 G. YING is B α ,n such that t B, so L is eventually in B Mα . ∈ B 0 ∈ ∩ 0 Thus, f(L) is eventually in f(B) α ,n (n, Λ ); hence, S is ∈ P 0 ⊂ P 0 cofinally in f(B) st(x, (n, Λ0)). So st(x, (n, Λ0)) is a sequential neighborhood of ⊂x by RemarkP 1.4(1). P (4) = (1). We can assume α,0 = Xα for each α Λ. For ⇒ P { } ∈ α Λ and n ω, we write α,n = Pβ : β Aα,n , where Aα,n ∈ ∈ P { ∈ } is countable. Put An = Aα,n : α Λ , then n = α,n : α ∪{ ∈ } P ∪{P ∈ Λ = Pβ : β An , and n : n ω is a sequence of point-finite } { ∈ } {P ∈ } covers of X. We can assume Aα,n : α Λ and An : n ω are { ∈ } { ∈ } all mutually disjoint. Let every An be endowed with the discrete topology. Put

M = b = (βn) Πn ωAn: there is α Λ such that Pβn α,n for { ∈ ∈ ∈ ∈ P each n ω, and Pβ is a network at some point xb in X . ∈ { n } } Then M, which is a subspace of Tychonoff product space Π An : n N , is a metric space. It is easy to check that f : M { X ∈ } −→ defined by f(b) = xb is a mapping. Claim 1. M is locally separable. Let b = (βn) M. Put Mb = c = (γn) M : γ0 = β0 . Then ∈ { ∈ } b Mb and Mb is open. Let α Λ such that Pβ α,0 = Xα . ∈ ∈ 0 ∈ P { } (i) Mb Πn ωAα,n: Let c = (γn) Mb. Then Pγ0 = Pβ0 = ⊂ ∈ ∈ Xα α,0. For each n ω, Pγ α,n = Pβ : β Aα,n , hence ∈ P ∈ n ∈ P { ∈ } γn Aα,n, so c Πn ωAα,n. ∈ ∈ ∈ (ii) Mb is separable: Notice that every Aα,n is a countable and discrete space; Πn ωAα,n is separable; so, Mb is separable. By (i),(ii) above,∈ M is locally separable. Claim 2. f is compact. Let x X and let Λ = α Λ: x Xα . Put Bα,n = ∈ 0 { ∈ ∈ } β Aα,n : x Pβ for each α Λ. Then Bα,n is finite; hence, K{ =∈ (Π ∈ B } ) is a compact∈ subset of M. We prove only α Λ0 n ω α,n 1 ∪ ∈ ∈ f − (x) = K. 1 (i) K f (x): Let b = (βn) K. Then there is α Λ such ⊂ − ∈ ∈ 0 that b = (βn) Πn ωBα,n. So βn Bα,n for each n ω, i.e., ∈ ∈ ∈ ∈ Pβn α,n, and x Pβn . Hence, Pβn is a network at x; thus, ∈ P ∈1 { } f(b) = x, i.e., b f − (x). 1 ∈ 1 (ii) f (x) K: Let b = (βn) f (x). Then f(b) = x; there − ⊂ ∈ − is α Λ such that Pβ α,n for each n N, and Pβ is a ∈ 0 n ∈ P ∈ { n } network at x. So βn Aα,n, and x Pβ , i.e., βn Bα,n; thus, ∈ ∈ n ∈ b = (βn) Πn ωBα,n K. ∈ ∈ ⊂ COMPACT IMAGES 357

1 By (i),(ii) above, f − (x) = K. Claim 3. f is pseudo-sequence-covering. Let x X, xn be a sequence in X converging to x. Put ∈ { } S = xn : n N x . There is a ﬁnite Λ Λ such that for { ∈ } ∪ { } 0 ⊂ each n ω, st(x, (n, Λ0)) is a sequential neighborhood of x. We can assume∈ (n, ΛP) covers S. For each n ω, put P 0 ∈

00 (n, Λ0) = Pβ : β Aα,n, α Λ0, (S st(x, (n, Λ0))) Pβ = φ , P { ∈ ∈ − P ∩ 6 }

0 (n, Λ0) = ( (n, Λ0))x 00 (n, Λ0). P P ∪ P Then 0 (n, Λ ) covering S is a ﬁnite subfamily of (n, Λ ). It is P 0 P 0 easy to check st(x, 0 (n, Λ )) = st(x, (n, Λ )). P 0 P 0 Put 0 (n, Λ ) = 0 : α Λ , where each 0 α,n. Put P 0 ∪{Pα,n ∈ 0} Pα,n ⊂ P 0 = Pβ : β A0 , then A0 is a ﬁnite subset of Aα,n. Pα,n { ∈ α,n} α,n 0 (i) For each β Aα,n, where α Λ0, n ω, we construct Sβ as follows: ∈ ∈ ∈ Put Sβ = S Pβ if x Pβ and Sβ = (S st(x, (n, Λ ))) Pβ ∩ ∈ − P 0 ∩ if x Pβ. Then Sβ is compact in X. 6∈ (ii) S = Sβ : β A0 , α Λ for each n ω: ∪{ ∈ α,n ∈ } ∈ Let y S. If there is α Λ and β A0 such that x, y Pβ, ∈ ∈ ∈ α,n { } ⊂ then y S Pβ = Sβ; If any α Λ and any β A0 , x, y Pβ, ∈ ∩ ∈ ∈ α,n { } 6⊂ since 0 (n, Λ ) covers S. Pick α Λ and β A0 such that y Pβ, P 0 ∈ ∈ α,n ∈ then y st(x, (n, Λ0)). Thus, y (S st(x, (n, Λ0))) Pβ = Sβ. 6∈ P 0 ∈ − P ∩ Put K = α Λ (Πn ωAα,n), then K is compact. Put L = b = ∪ ∈ 0 ∈ { (βn) K : n ωSβn = φ . (iii)∈ L is∩ a closed∈ subset6 } of K; hence, L is compact: Let a = (αn) K L. Then there is α0 Λ0 such that 0 ∈ − ∈ a Πn ωAα ,n, and n ωSαn = φ. So there is n0 ω such that ∈ ∈ 0 ∩ ∈ ∈ n n0 Sαn = φ. Put W = b = (βn) K : βn = αn for n n0 . Then∩ ≤ W is an open neighborhood{ of x∈in K. We claim W ≤L = φ}; ∩ thus, L is closed subset of K. If not, let c = (γn) W L. Then ∈ ∩ n ωSγn n n0 Sγn = n n0 Sαn = φ by c W , and n ωSγn = φ by∩ ∈c L.⊂ This ∩ ≤ is a contradiction.∩ ≤ ∈ ∩ ∈ 6 (iv)∈ L M and f(L) S: ⊂ ⊂ Let b = (βn) L. Then n ωSβn = φ, and there is α Λ0 ∈ ∈0 ∩ 6 ∈ such that βn Aα,n Aα,n for any n ω. Pick y n ωSβn ∈ ⊂ ∈ ∈ ∩ ∈ ⊂ 358 G. YING

n ωPβn , then y Pβn α,n for any n ω. So Pβn is a network ∩ ∈ ∈ ∈ P ∈ { } at y; thus, b M and f(b) = y Sβn S. (v) S f(∈L): ∈ ⊂ ⊂ For limit point x of S, pick α Λ such that x Xα. Then for ∈ 0 ∈ any n ω, there is αn A0 such that x Pα . So x S Pα = ∈ ∈ α,n ∈ n ∈ ∩ n Sαn ; hence, n ωSαn = φ. It is easy to see that a = (αn) L and f(a) = x. For∩ y∈ S and6 y = x, by (a), there is n ω such∈ that y ∈ 6 ∈ 6∈ st(x, (n, Λ0)). Put m = min n : y st(x, (n, Λ0)) . Then y P { 6∈ P 0 } ∈ st(x, (m 1, Λ0)), so there is α Λ0 and βm 1 Aα,m 1 such that P − ∈ − ∈ − x, y Pβm 1 . Thus, y S Pβm 1 = Sβm 1 . For n < m 1, as { } ⊂ − ∈ ∩ − − − α,m 1 refines α,n, there is βn Aα,n such that Pβm 1 Pβn . So P − P ∈ − ⊂ x, y Pβ ; thus, βn A0 and y S Pβ = Sβ . For n m, as { } ⊂ n ∈ α,n ∈ ∩ n n ≥ α,n refines α,m 1, y st(x, (n, Λ0)), i.e., y S st(x, (n, Λ0)). P P − 6∈ P ∈ − P Notice that y Xα, pick βn Aα,n such that y Pβ . Obviously, ∈ ∈ ∈ n βn A0 and y (S st(x, (n, Λ ))) Pβ = Sβ . It is easy to ∈ α,n ∈ − P 0 ∩ n n see that b = (βn) L and f(b) = y. So S f(L). By (i)-(v) above,∈ f is pseudo-sequence-covering.⊂ By the above, X is a pseudo-sequence-covering compact image of a locally separable metric space. Corollary 2.3. For a space X, the following are equivalent. (1) X is a pseudo-sequence-covering, quotient compact image of a locally separable metric space; (2) X is a quotient compact image of a locally separable metric space. Proof: (1) = (2) is obvious; we prove only (2) = (1). By [7, Proposition⇒ 2.1.16(5)], quotient mappings⇒ on sequential spaces are sequentially-quotient, so X is a sequentially-quotient compact image of a locally separable metric space; hence, X is a pseudo-sequence-covering compact image of a locally separable metric space from Theorem 2.2. Quotient mappings preserve sequential spaces, so X is sequential. By [7, Proposition 2.1.16(2)], X is a pseudo-sequence-covering, quotient compact image of a locally separable metric space. Remark 2.4. In [5], Ikeda asked if every quotient compact image of a locally separable metric space must be the pseudo-sequence- covering, quotient compact images of a locally separable metric space. Corollary 2.3 answers this question affirmatively. COMPACT IMAGES 359

Remark 2.5. By [17, Theorem 1], Theorem 2.2, and [10, Theo- rem 4.6], sequentially-quotient compact images of metric spaces (locally separable metric spaces, separable metric spaces) are pseudo- sequence-covering compact images of metric spaces (locally separable metric spaces, separable metric spaces). But we don’t even know whether sequentially-quotient compact mappings on separable metric spaces are pseudo-sequence-covering. So we raised the following question. By the above, we conjecture the answer is pos- itive. Question 2.6. Let f : X Y be a sequentially-quotient compact mapping from a separable−→ metric spaces X onto a space Y . Is f pseudo-sequence-covering? Acknowledgment. The author would like to thank the referee for his valuable amendments. References 1. P. S. Alexandroff, On some results concerning topological spaces and their continuous mappings. 1962 General Topology and its Relations to Mod- ern Analysis and Algebra (Proc. Sympos., Prague, 1961) pp. 41–54. NY: Academic Press. 2. J. R. Boone and F. Siwiec, Sequentially quotient mappings, Czechoslovak Math. J. 26(101) (1976), no. 2, 174–182. 3. R. Engelking, General Topology. Warszawa: Polish Scientific Publishers, 1977. 4. G. Gruenhage, E. Michael, and Y. Tanaka, Spaces determined by point- countable cover, Pacific J. Math. 113 (1984), 303–332. 5. Y. Ikeda, σ-strong networks and quotient compact images of metric spaces, Questions Answers Gen. Topology 17 (1999), 269-279. 6. Y. Ikeda and Y. Tanaka, Spaces having star-countable k-networks, Topology Proc. 18 (1993), 107–132. 7. S. Lin, Generalized Metric Spaces and Mappings. (Chinese). Beijing: Chi- nese Science Press, 1995. 8. S. Lin, Point-Countable Covers and Sequence-Covering Mappings. (Chi- nese). Beijing: Chinese Science Press, 2002. 9. S. Lin., C. Liu, and M. Dai, Images on locally separable metric spaces, Acta Math. Sinica (N. S.) 13 (1997), 1–8. 10. S. Lin and P. Yan, Sequence-covering maps of metric spaces, Topology Appl. 109 (2001), 301–314. 11. S. Lin and P. Yan, On sequence-covering compact mappings, (Chinese), Acta Math. Sinica 44 (2001), 175–182. 12. E. Michael and K. Nagami, Compact-covering images of metric spaces, Proc. Amer. Math. Soc. 37 (1973), 260–266. 360 G. YING

13. F. Siwiec, Sequence-covering and countably bi-quotient mappings, General Topology and Appl. 1 (1971), 143–154. 14. Y. Tanaka, Point-countable covers and k-networks, Topology Proc. 12 (1987), 327–349. 15. Y. Tanaka, Theory of k-networks, II, Questions Answers Gen. Topology 19 (2001), 27–46. 16. Y. Tanaka, C. Liu, and Y. Ikeda, Around quotient compact images of metric spaces. To appear in Topology Appl. 17. P. Yan, Compact images of metric spaces, (Chinese), J. Math. Study 30 (1997), no. 2, 185–187.

Department of Mathematics, Suzhou University, Suzhou 215006, P. R. China E-mail address: [email protected]