2_Dung.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 2, No. 2, 2009, (182-194) ISSN 1307-5543 – www.ejpam.com On 1-Sequence-Covering π-s-Images of Locally Sep- arable Metric Spaces Nguyen Van Dung Mathematics Faculty, Dongthap University Caolanh City, Dongthap Province, Vietnam Abstract. In this paper, we give a characterization on 1-sequence-covering π-s-images of lo- cally separable metric spaces by means of point-countable σ-strong sn-network consisting of cosmic spaces (sn-second countable spaces, ℵ0-spaces). As an application, we get a new char- acterization on 1-sequence-covering, quotient π-s-images of locally separable metric spaces, which is helpful in solving Y. Tanaka and S. Xia’s question in [21]. AMS subject classifications: 54D65, 54E35, 54E40 Key words: 1-sequence-covering, σ-strong sn-network, π-s-mapping 1. Introduction To determine what spaces the images of ŞniceŤ spaces under ŞniceŤ mappings are is one of the central questions of general topology [1]. In the past, many noteworthy results on images of metric spaces have been obtained. For a survey in this field, Email addresses: nvdung�staff.dthu.edu.vn; nguyendungt �yahoo. om http://www.ejpam.com 182 c© 2009 EJPAM All rights reserved. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 183 see [19], for example. Related to characterizations on images of metric spaces, Y. Tanaka and S. Xia posed the following question in [21]. Question 1.1 ( [21]). What is a nice characterization for a quotient s-image of a locally separable metric space? This question was partly answered by many authors [13], [14], [20]. It is known that 1-sequence-covering s-images of metric spaces have been characterized by point- countable sn-networks, and 1-sequence-coveringπ-images of metric spaces have been characterized byσ-strong sn-networks [16]. Also, as in the proofs of [10], 1-sequence- covering π-s-images of metric spaces can be characterized by point-countable σ- strong sn-networks. Recently, the characterizations on images of locally separable metric spaces cause attention once again, and 1-sequence-covering s-images of lo- cally separable metric spaces have been characterized by point-countable sn-network consisting of cosmic spaces (ℵ0-spaces) [5]. Taking these results into account, it is natural to be interested in the following question. Question 1.2. Are the following equivalent for a space X? 1. X is an 1-sequence-covering π-s-image of a locally separable metric space. 2. X has a point-countable σ-strong sn-network consisting of cosmic spaces (ℵ0- spaces). In this paper, we give a characterization on 1-sequence-covering π-s-images of lo- cally separable metric spaces by means of point-countable σ-strong sn-network con- sisting of cosmic spaces (sn-second countable spaces, ℵ0-spaces). As an application, we get a new characterization on 1-sequence-covering, quotient π-s-images of lo- cally separable metric spaces, which is helpful in solving the above Question 1.1 of Y. Tanaka and S. Xia. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 184 Throughout this paper, all spaces are regular and T1, N denotes the set of all natural numbers, ω = N ∪ {0}, and a convergent sequence includes its limit point. Let P be a family of subsets of X and x ∈ X . Then ⋂ P , and st(x ,P ) denote the intersection ⋂ {P : P ∈ P }, and the union ⋃ {P ∈ P : x ∈ P}, respectively. A convergent sequence {xn : n ∈ω} converging to x0 is eventually in a subset A of X , if {xn : n ≥ n0} ∪ {x0} ⊂ A for some n0 ∈ N. For terms which are not defined here, please refer to [3]. 2. Main Results Definition 2.1. Let P be a subset of a space X . (1) P is a sequential neighborhood of x [4], if for every convergent sequence S converging to x in X , S is eventually in P. (2) P is a sequentially open subset of X [4], if for every x ∈ P, P is a sequential neighborhood of x. Definition 2.2. Let P be a family of subsets of a space X . (1) For each x ∈ X , P is a network at x in X , if x ∈ ⋂ P , and if x ∈ U with U open in X , there exists P ∈ P such that x ∈ P ⊂ U. (2) P is a cs-network of X [8], if for every convergent sequence S converging to x ∈ U with U open in X , there exists P ∈ P such that S is eventually in P ⊂ U. (3) P is an sn-cover of X [14], if each element of P is a sequential neighborhood of some point in X , and for each x ∈ X , some P ∈ P is a sequential neighborhood of x. Definition 2.3. Let X be a space. (1) X is an ℵ0-space [17] (resp., cosmic space [17], sn-second countable space [7]), if X has a countable cs-network (resp., countable network, countable sn-network). (2) X is a sequential space [4], if every sequentially open subset of X is open. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 185 (3) X is sequentially separable [2], if X has a countable subset D such that for each x ∈ X , there exists a sequence L ⊂ D converging to x, where D is a sequentially dense subset of X . Definition 2.4. Let P = ⋃ {Px : x ∈ X} be a family of subsets of a space X satisfying that, for each x ∈ X , Px is a network at x in X , and if U , V ∈ Px , then W ⊂ U ∩ V for some W ∈ Px . (1) P is a weak base of X [18], if G ⊂ X such that for each x ∈ G, there exists P ∈ Px satisfying P ⊂ G, then G is open in X (2)P is an sn-network of X [12], if each member ofPx is a sequential neighborhood of x in X . (3) The above Px is respectively a weak base, and an sn-network at x in X [11]. Remark 2.5 ( [14]). An sn-network of a sequential space is a weak base. Definition 2.6. Let f : X −→ Y be a mapping. (1) f is an 1-sequence-covering mapping [12], if for every y ∈ Y , there exists x y ∈ f −1(y) such that whenever {yn : n ∈ N} is a sequence converging to y in Y there exists a sequence {xn : n ∈ N} converging to x y in X with each xn ∈ f −1(yn). (2) f is an 1-sequentially quotient mapping [16], if for every y ∈ Y , there exists x y ∈ f −1(y) such that whenever {yn : n ∈ N} is a sequence converging to y in Y there exists a sequence {xk : k ∈ N} converging to x y in X with each xk ∈ f −1(ynk ). (3) f is a π-mapping [1], if for every y ∈ Y and for every neighborhood U of y in Y , d( f −1(y), X − f −1(U))> 0, where X is a metric space with a metric d. (4) f is an s-mapping [1], if f −1(y) is separable for every y ∈ Y . (5) f is a π-s-mapping [10], if f is both π-mapping and s-mapping. Definition 2.7. Let {Pn : n ∈ N} be a refinement sequence of a space X , i.e., each Pn is a cover of X and Pn+1 is a refinement of Pn. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 186 (1) ⋃ {Pn : n ∈ N} is σ-strong network of X [9], if {st(x ,Pn) : n ∈ N} is a network at x in X for every x ∈ X . (2) ⋃ {Pn : n ∈ N} is weak development of X [10], if {st(x ,Pn) : n ∈ N} is a weak base at x in X for every x ∈ X . (3) ⋃ {Pn : n ∈ N} is a σ-strong sn-network of X , if ⋃ {Pn : n ∈ N} is a σ-strong network and each Pn is an sn-cover of X . A σ-strong sn-network of X is a point-star network of sn-covers in the sense of [16]. (4) ⋃ {Pn : n ∈ N} is an sn-weak-development of X , if ⋃ {Pn : n ∈ N} is a weak- development and each Pn is an sn-cover of X . Definition 2.8. Let P = ⋃ {Pn : n ∈ N} be a σ-strong network of X . For every n ∈ N, put Pn = {Pα : α ∈ An}, and endowed An with discrete topology. Put M = n a = (αn) ∈ ∏ n∈N An : {Pαn : n ∈ N} forms a network at some point xa in X o . Then M, which is a subspace of the product space ∏ n∈N An, is a metric space, xa is unique, and xa = ⋂ n∈N Pαn for every a ∈ M. Define f : M → X by choosing f (a) = xa, then f is a mapping and ( f , M , X , {Pn}) is a Ponomarev-system [15]. Theorem 2.9. The following are equivalent for a space X . 1. X is an 1-sequence-covering π-s-image of a locally separable metric space. 2. X is an 1-sequentially-quotient π-s-image of a locally separable metric space. 3. X has a point-countable σ-strong sn-network consisting of sn-second countable spaces. 4. X has a point-countable σ-strong sn-network consisting of ℵ0-spaces. 5. X has a point-countable σ-strong sn-network consisting of cosmic spaces. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 187 Proof. (1)⇒ (2). It is obvious. (2) ⇒ (3). Let f : M −→ X be an 1-sequentially-quotient π-s-mapping from a locally separable metric space M with a metric d onto X . For each x ∈ X , there exists ax ∈ f −1(x) such that whenever {xn : n ∈ N} is a sequence converging to x in X there exists a sequence {ak : k ∈ N} converging to ax in M with each ak ∈ f −1(xnk ). Since M is locally separable metric, M = ⊕λ∈ΛMλ by [3, 4.4.F], where each Mλ is a separable metric space with a metric dλ. For each λ ∈ Λ, let Dλ be a countable dense subset of Mλ. For each n ∈ N, put Bλ,n,x = {B(a, 1/n) : a ∈ Dλ, ax ∈ B(a, 1/n)}, where B(a, 1/n) = {b ∈ Mλ : dλ(a, b) < 1/n}, and put Bn,x = ⋃ {Bλ,n,x : λ ∈ Λ},Bn = ⋃ {Bn,x : x ∈ X},Bx = ⋃ {Bn,x : n ∈ N}, B = ⋃ {Bn : n ∈ N}= ⋃ {Bx : x ∈ X}, and Pn,x = f (Bn,x),Pn = ⋃ {Pn,x : x ∈ X},Px = ⋃ {Pn,x : n ∈ N}, P = ⋃ {Pn : n ∈ N}= ⋃ {Px : x ∈ X}. Then {Pn : n ∈ N} is a refinement sequence of X . We shall prove that P is a point- countable σ-strong sn-network of X consisting of sn-second countable spaces by the following facts (a), (b), (c), and (d). (a) ⋃ {Pn : n ∈ N} is a σ-strong network of X . Let x ∈ U with U open in X . Since f is a π-mapping, d( f −1(x), M − f −1(U))> 0. It implies that d( f −1(x), M − f −1(U)) > 2/n for some n ∈ N. Let x ∈ f (B(a, 1/n)) ∈ Pn for some B(a, 1/n) ∈ Bλ,n,x . We shall prove that B(a, 1/n) ⊂ f −1(U). In fact, if B(a, 1/n) 6⊂ f −1(U), then there exists b ∈ B(a, 1/n) − f −1(U). Since f −1(x) ∩ N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 188 B(a, 1/n) 6= ;, there exists c ∈ f −1(x) ∩ B(a, 1/n). Then d( f −1(x), M − f −1(U)) ≤ d(c, b) ≤ d(c, a)+ d(a, b) < 2/n. It is a contradiction. Then we get f (B(a, 1/n)) ⊂ U . Therefore, st(x ,Pn) = ⋃ { f (B(a, 1/n)) : x ∈ f (B(a, 1/n)), a ∈ Dλ,λ ∈ Λ} ⊂ U . It implies that P is a σ-strong network of X . (b) Each Pn is an sn-cover of X . Let x ∈ X and P = f (B) ∈ Pn,x for some B ∈ Bn,x . We shall prove that P is a sequential neighborhood of x . Let S be a convergent sequence converging to x in X . Then there exists a convergent sequence L converging to ax in M such that f (L) is a subsequence of S. Since B is open, L is eventually in B. Hence f (Lλ) is eventually in P. It implies that S is frequently in P. It follows from [6, Remark 1.4] that P is a sequential neighborhood of x . Therefore, Pn is an sn-cover of X . (c) P is point-countable. Let x ∈ X . Since f is an s-mapping, f −1(x) is separable. It implies that f −1(x) meets at most countably many Mλ’s. Then f −1(x) meets at most countably many members of Bn, i.e., x meets at most countable many members of Pn. Therefore, P is point-countable. (d) Each P ∈ P is an sn-second countable space. Let P = f (B) for some B ∈ B . Since B is separable metric, P is sequentially separable by [14, Lemma 2.2]. Let DP be a sequentially dense subset of P. For each x ∈ P, put Qx = {Q ∩ P : Q ∈ Px ,Q ∩ DP 6= ;}, and put Q = ⋃ {Qx : x ∈ P}. Since P is point-countable and DP is countable, Q is countable. It suffices to prove the following facts (i), (ii), and (iv) for every x ∈ P. (i) Qx is a network at x in P. Let x ∈ U with U open in P. Then x ∈ V with V open in X and V ∩ P = U . Let S be a sequence in DP converging to x . Since P is a σ-strong sn-network of X , S ∪ {x} is eventually in Q ⊂ V with some Q ∈ Px . It implies that Q ∩ DP 6= ;, and x ∈ Q ∩ P ⊂ V ∩ P = U . Therefore,Qx is a network at x in P. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 189 (ii) If Q1,Q2 ∈ Qx , then Q ⊂Q1 ∩Q2 for some Q ∈ Qx . Let Q1 = f (B1)∩ P,Q2 = f (B2)∩ P for some B1, B2 ∈Bx . Let S be a sequence in DP converging to x . Then there exists a sequence L converging to ax in M such that f (L) is a subsequence of S. Since Bx is a base at ax in M , there exists C ∈ Bx such that L ∪ {ax} is eventually in C ⊂ B1 ∩ B2. Then S ∪ {x} is frequently in f (C). It implies that f (C)∩ DP 6= ;. Put Q = f (C)∩ P. Then Q ∈ Qx , and Q ⊂Q1 ∩Q2. (iii) Each Q ∈ Qx is a sequential neighborhood of x in P. Let Q = f (C)∩ P with some C ∈Bx , and f (C)∩DP 6= ;, and let S be a convergent sequence converging to x in P. Then there exists a convergent sequence L converging to ax in M such that f (L) is a subsequence of S. Since L is eventually in C , S is frequently in Q. It follows from [6, Remark 1.4] that Q is a sequential neighborhood of x in P. (3)⇒ (4)⇒ (5). It is obvious. (5) ⇒ (1). Let P = ⋃ {Pn : n ∈ N} be a point-countable σ-strong sn-network of X consisting of cosmic spaces. For each n ∈ N, put Pn = {Pn,λ : λ ∈ Λn} = ⋃ {Pn,x : x ∈ X}, where each Pn,x is an sn-cover at x in X . Since each Pn,λ is a cosmic space, Pn,λ is a sequentially separable space by [14, Corollary 2.6]. Then Pn,λ has a countable sequentially dense subset Dn,λ. For each i ∈ N and x ∈ Pn,λ, put Qn,λ,i,x = {P∩ Pn,λ : P ∈ Pi,x , P∩Dn,λ 6= ;}, and putQn,λ,i = ⋃ {Qn,λ,i,x : x ∈ Pn,λ}, and Qn,λ = ⋃ {Qn,λ,i : i ∈ N}. Since P is a point-countable and Dn,λ is countable, Qn,λ is a countable. It is easy to see that {Qn,λ,i : i ∈ N} is a refinement sequence of Pn,λ. For each x ∈ U with U open in Pn,λ, we get x ∈ V with V open in X and V∩Pn,λ = U . Since P is a σ-strong network of X , there exists i ∈ N such that x ∈ st(x ,Pi)⊂ V . Let L be a sequence in Dn,λ converging to x . Since Pi is an sn-cover of X , L∪{x} is eventually in P ⊂ V for some P ∈ Pi,x . Then P ∩ Dn,λ 6= ;, and P ∩ Pn,λ ∈ Qn,λ,i. It implies that x ∈ st(x ,Qn,λ,i) = st(x ,Pi) ∩ Pn,λ ⊂ V ∩ Pn,λ = U . Therefore, {Qn,λ,i : i ∈ N} is a σ-strong network of Pn,λ. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 190 For each x ∈ Pn,λ and i ∈ N, let Q ∈ Qn,λ,i,x . Then Q = P ∩ Pn,λ, where P ∈ Pi,x and P ∩ Dn,λ 6= ;. Let S be a convergent sequence converging to x in Pn,λ. Since Pi is an sn-cover of X , S is eventually in P. It implies that S is eventually in P ∩ Pn,λ. Therefore, Qn,λ is a σ-strong sn-network of Pn,λ. By the above, the Ponomarev-system ( fn,λ, Mn,λ, Pn,λ, {Qn,λ,i}) exists. Since each Qn,λ,i is countable, Mn,λ is a separable metric space with the metric dn,λ described as follows. For a = (αi), b = (βi) ∈ Mn,λ, if a = b, then dn,λ(a, b) = 0, and otherwise, dn,λ(a, b) = 1/min{i ∈ N : αi 6= βi}. Put M = ⊕{Mn,λ : λ ∈ Λn, n ∈ N} and define f : M −→ X by choosing f (a) = fn,λ(a) if a ∈ Mn,λ with λ ∈ Λn, n ∈ N. Then f is a mapping, and M is a locally separable metric space with the metric d described as follows. For each a, b ∈ M , if a, b ∈ Mn,λ for some λ ∈ Λn and n ∈ N, then d(a, b) = dn,λ(a, b), and otherwise, d(a, b) = 1. We shall prove that f is a sequence- covering π-s-mapping by the following facts (a), (b), and (c). (a) f is a π-mapping. Let x ∈ U with U open in X . Then st(x ,Pm) ⊂ U for some m ∈ N. For each n ∈ N and λ ∈ Λn with x ∈ Pn,λ, we get st(x ,Qn,λ,m)⊂ Un,λ, where Un,λ = U ∩ Pn,λ. For each a = (αi) ∈ Mn,λ, if dn,λ( f −1 n,λ (x), a) < 1/m, then there exists b = (βi) ∈ f −1 n,λ (x) such that dn,λ(a, b) < 1/m. Hence αi = βi if i ≤ m. Since x ∈ Qβm ⊂ st(x ,Qn,λ,m) ⊂ Un,λ, fn,λ(a) ∈ Qαm = Qβm ⊂ Un,λ. It implies that a ∈ f −1 n,λ (Un,λ). Therefore, if a ∈ Mn,λ − f −1 n,λ (Un,λ), then dn,λ( f −1 n,λ (x), a) ≥ 1/m. Hence dn,λ( f −1 n,λ (x), Mn,λ − f −1 n,λ (Un,λ)) ≥ 1/m. So we get d( f −1(x), M − f −1(U)) = inf � d(a, b) : a ∈ f −1(x), b ∈ M − f −1(U) =min n 1, inf � dn,λ( f −1 n,λ (x), Mn,λ− f −1 n,λ (Un,λ)) : λ ∈ Λn, n ∈ N o ≥ 1/m > 0. It implies that f is a π-mapping. (b) f is an s-mapping. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 191 Let x ∈ X . For each n ∈ N, since Pn is point-countable, Λn,x = {λ ∈ Λn : x ∈ Pn,λ} is countable. Since each Mn,λ is separable metric, f −1 n,λ (x) is separable. It implies that f −1(x) = ⋃ { f −1 n,λ (x) : λ ∈ Λn,x , n ∈ N} is separable. Then f is an s-mapping. (c) f is 1-sequence-covering. For each x ∈ X , since P is a σ-strong sn-network of X , there exists Pn,λ ∈ P such that for each convergent sequence S converging to x in X , S is eventually in Pn,λ. Since Qn,λ is a countable σ-strong sn-network of Pn,λ, fn,λ is an 1-sequence- covering mapping as in the proof (3) ⇒ (1) of [16, Theorem 11]. Then there exists ax ∈ Mn,λ such that whenever Hn,λ is a convergent sequence converging to x in Pn,λ there exists a convergent sequence Kn,λ converging to ax in Mn,λ with fn,λ(Kn,λ) = Hn,λ. Put Hn,λ = S ∩ Pn,λ, then Hn,λ is a convergent sequence converging to x in Pn,λ. Since S− Pn,λ is finite, S− Pn,λ = f (F) for some finite subset F of M . Put K = Kn,λ ∪ F , then K is a convergent sequence converging to ax in M satisfying f (K) = S. It implies that f is 1-sequence-covering. Corollary 2.10. The following are equivalent for a space X . 1. X is an 1-sequence-covering, quotient π-s-image of a locally separable metric space. 2. X is an 1-sequentially-quotient, quotient π-s-image of a locally separable metric space. 3. X has a point-countable sn-weak-development consisting of sn-second countable spaces. 4. X has a point-countable sn-weak-development consisting of ℵ0-spaces. 5. X has a point-countable sn-weak-development consisting of cosmic spaces. Proof. (1)⇒ (2). It is obvious. N. Dung / Eur. J. Pure Appl. Math, 2 (2009), (182-194) 192 (2)⇒ (3). By Theorem 2.9, X has a point-countable σ-strong sn-network ⋃ {Pn : n ∈ N} consisting of sn-second countable spaces. We shall prove that, for every x ∈ X , {st(x ,Pn) : n ∈ N} is a weak base at x in X by the following facts (a), (b), and (c). (a) {st(x ,Pn) : n ∈ N} is a network at x in X . It follows from the fact that ⋃ {Pn : n ∈ N} is a σ-strong network of X . (b) If st(x ,Pk), st(x ,Pl) ∈ {st(x ,Pn) : n ∈ N}, then there exists m ∈ N such that st(x ,Pm) ⊂ st(x ,Pk)∩ st(x ,Pl). It is clear by choosing m =max{k, l}. (c) Since X is a quotient image of a metric space, X is sequential. It is easy to see that each st(x ,Pn) is a sequential neighborhood of x in X . Note that {st(x ,Pn) : n ∈ N} is a network at x in X . Then {st(x ,Pn) : n ∈ N} is an sn-network at x in X . It implies that {st(x ,Pn) : n ∈ N} is a weak base at x by Remark 2.5. (3)⇒ (4)⇒ (5). It is obvious. (5) ⇒ (1). Let P = ⋃ {Pn : n ∈ N} be a point-countable sn-weak-development of X consisting of cosmic spaces. By Theorem 2.9, X is an 1-sequence-covering π- s-image of a locally separable metric space under the mapping f . Since X has a weak-development, X is sequential. In fact, let A be a sequentially open subset of X and x ∈ A. For each n ∈ N, if st(x ,Pn) 6⊂ A, then there exists xn ∈ st(x ,Pn) − A. We get that {xn : n ∈ N} is a sequence converging to x . Then {xn : n ∈ N} ∪ {x} is eventually in A. It is a contradiction. Therefore, st(x ,Pn)⊂ A for some n ∈ N. Hence A is open, i.e., X is sequential. Since X is sequential, f is quotient by [14, Lemma 3.5]. It implies that X is an 1-sequence-covering, quotient π-s-image of a locally separable metric space. Remark 2.11. Corollary 2.10 is a partly answer of Question 1.1. REFERENCES 193 References [1] A. V. Arhangel’skii, Mappings and spaces, Russian Math. Surveys 21: 115 – 162 (1966). [2] S. W. Davis, More on Cauchy conditions, Topology Proc. 9: 31 – 36 (1984). [3] R. Engelking, General topology, Sigma series in pure mathematics, vol. 6, Heldermann Verlag, Berlin, 1988. [4] S. P. Franklin, Spaces in which sequences suffice, Fund. Math. 57: 107 – 115 (1965). [5] Y. Ge, 2-sequence-covering mappings in Ponomarev-systems, Chinese Adv. Math. (China), to appear. [6] Y. Ge, On compact images of locally separable metric spaces, Topology Proc. 27, 1: 351 – 360 (2003). [7] Y. Ge, Spaces with countable sn-networks, Comment. Math. Univ. Carolina 45: 169 – 176 (2004). [8] J. A. Gurthrie, A characterization of ℵ0-spaces, General Topology Appl. 1: 105 – 110 (1971). [9] Y. Ikeda, C. Liu, and Y. Tanaka, Quotient compact images of metric spaces, and related matters, Topology Appl. 122, 237 – 252 (2002). [10] Z. Li, On π-s-images of metric spaces, Int. J. Math. Sci. 7: 1101 – 1107 (2005). [11] Z. Li, Q. Li, and X. Zhou, On sequence-covering msss-maps, Mat. Vesnik 59: 15 – 21 (2007). [12] S. Lin, On sequence-covering s-mappings, Adv. Math. (China) 25: 548 – 551 (1996). [13] S. Lin, C. Liu, and M. Dai, Images on locally separable metric spaces, Acta Math. Sinica (N.S.) 13, 1: 1 – 8 (1997). [14] S. Lin and P. Yan, Sequence-covering maps of metric spaces, Topology Appl. 109: 301 – 314 (2001). [15] S. Lin and P. Yan, Notes on c f p-covers, Comment. Math. Univ. Carolina 44, 2: 295 – 306 (2003). [16] S. Lin, J-C. Zhu, Y. Ge, and J-S. Gu, Almost-open maps, sequence-covering maps and sn-networks, Indian J. Pure Appl. Math. 37, 2: 111 – 119 (2006). REFERENCES 194 [17] E. Michael, ℵ0-spaces, J. Math. Mech. 15: 983–1002 (1966). [18] F. Siwiec, On defining a space by a weak-base, Pacific J. Math. 52: 233 – 245 (1974). [19] Y. Tanaka, Theory of k-networks II, Questions Answers in Gen. Topology 19: 27 – 46 (2001). [20] Y. Tanaka and Y. Ge, Around quotient compact images of metric spaces, and symmetric spaces, Houston J. Math. 32, 1: 99 – 117 (2006). [21] Y. Tanaka and S. Xia, Certain s-images of locally separable metric spaces, Questions Answers Gen. Topology 14: 217 – 231 (1996). [22] P. Yan, On strong sequence-covering compact mappings, Northeast. Math. J. 14: 341 – 344 (1998).