PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 87(101) (2010), 143–153 DOI: 10.2298/PIM1001143V

ON SEQUENCE-COVERING mssc-IMAGES OF LOCALLY SEPARABLE METRIC SPACES

Nguyen Van Dung

Communicated by Miloš Kurilić

Abstract. We characterize sequence-covering (resp., 1-sequence-covering, 2-sequence-covering) mssc-images of locally separable metric spaces by means of σ-locally finite cs-networks (resp., sn-networks, so-networks) consisting of ℵ_{0}-spaces (resp., sn-second countable spaces, so-second countable spaces). As the applications, we get characterizations of certain sequence-covering, quotient mssc-images of locally separable metric spaces.

1. Introduction

A study of some images of metric spaces under certain mappings is an important task on general topology. In [12], Li characterized sequence-covering (pseudosequence-covering) mssc-images of metric spaces by means of ℵ-spaces as follows.

Theorem 1.1. [12, Theorem 4] The following are equivalent for a space X.

(1) X is an ℵ-space.

(2) X is a sequence-covering mssc-image of a metric space.

(3) X is a pseudo-sequence-covering mssc-image of a metric space.

In [18], Lin and Yan characterized compact-covering, quotient π- and msscimages of metric spaces by means of g-metrizable spaces, and this result has been proved by a quick and systematic proof in [25].

Theorem 1.2. [18, Corollary 18] The following are equivalent for a space X.

(1) X is a g-metrizable space.

(2) X is a compact-covering, quotient compact and mssc-image of a metric space.

(3) X is a compact-covering, quotient π- and mssc-image of a metric space.

(4) X is a compact-covering, quotient π- and σ-image of a metric space.

2010 Mathematics Subject Classification: Primary 54E35, 54E40; Secondary 54D55, 54E99.

Key words and phrases: mssc-mapping, sequence-covering, 1-sequence-covering, 2-sequence-covering, σ-locally finite, cs-network, sn-network, so-network.

Related to the characterizations of images of metric spaces, many topologists were engaged in characterizing images of locally separable metric spaces, and some noteworthy results have been shown. In [16], Lin, Liu, and Dai characterized quotient s-images of locally separable metric spaces. After that, Lin and Yan characterized sequence-covering s-images of locally separable metric spaces [17]; Ikeda, Liu and Tanaka characterized quotient compact images of locally separable metric spaces [11]; Ge characterized pseudo-sequence-covering compact images of locally separable metric spaces [8]; An and Dung characterized quotient π-images of locally separable metric spaces [1]. In general, it is difficult to obtain nice characterizations of images of locally separable metric spaces (under covering-mappings) instead of metric domains.

Take the above into account, note that ℵ-spaces and g-metrizable spaces are spaces having certain σ-locally finite networks, the following question arises naturally.

Question. How are sequence-covering (1-sequence-covering, 2-sequence-cover-ing) mssc-images of locally separable metric spaces characterized by means of σlocally finite networks?

In this paper, we characterize sequence-covering (resp., 1-sequence-covering, 2-sequence-covering) mssc-images of locally separable metric spaces by means of σlocally finite cs-networks (resp., sn-networks, so-networks) consisting of ℵ_{0}-spaces (resp., sn-second countable spaces, so-second countable spaces). As the applications, we get characterizations of certain sequence-covering, quotient mssc-images of locally separable metric spaces. These results make the study of images of locally separable metric spaces more completely.

Throughout this paper, all spaces are regular and T_{1}, all mappings are continuous and onto, a convergent sequence includes its limit point, and ℕ denotes the set of all natural numbers. Let f : X → Y be a mapping, and P be a family of subsets of X, we denote ⋃ P = ⋃ {P : P ∈ P}, ⋂ P = ⋂ {P : P ∈ P}, and f(P) = {f(P) : P ∈ P}. We say that a convergent sequence {x_{n} : n ∈ ℕ} ∪ {x} converging to x is eventually in A if {x_{n} : n ⩾ n_{0}} ∪ {x} ⊂ A for some n_{0} ∈ ℕ, and it is frequently in A if {x_{n_{k}} : k ∈ ℕ} ∪ {x} ⊂ A for some subsequence {x_{n_{k}} : k ∈ ℕ} of {x_{n} : n ∈ ℕ}.

Definition 1.1. Let P be a family of subsets of a space X.

(1) P is a network for X [19] if, P = ⋃ {P_{x} : x ∈ X}, where x ∈ ⋂ P_{x}, and if x ∈ U with U open in X, then there exists P ∈ P_{x} such that x ∈ P ⊂ U for every x ∈ X. Here, P_{x} is a network at x in X.

(2) P is a cs-network for X [10] if, for each convergent sequence S converging to x ∈ U with U open in X, S is eventually in P ⊂ U for some P ∈ P.

(3) P is a cs^{∗}-network for X [7] if, for each convergent sequence S converging to x ∈ U with U open in X, S is frequently in P ⊂ U for some P ∈ P.

(4) P is a cfp-network for X [26] if, for each compact subset H ⊂ U with U open in X, there exists a finite subfamily F of P such that H ⊂ ⋃ {C_{F} : F ∈ F} ⊂ U, where C_{F} is closed and C_{F} ⊂ F for every F ∈ F.

Definition 1.2. [6] Let X be a space and P be a subset of X.

(1) P is a sequential neighborhood of x in X, if whenever S is a convergent sequence converging to x, then S is eventually in P.

(2) P is a sequentially open subset of X, if P is a sequential neighborhood of x in X for every x ∈ P.

Definition 1.3. Let P = ⋃ {P_{x} : x ∈ X} be a family of subsets of a space X satisfying that, for each x ∈ X, P_{x} is a network at x in X, and if U, V ∈ P_{x}, then W ⊂ U ∩ V for some W ∈ P_{x}.

(1) P is a weak base for X [23], if G ⊂ X such that for each x ∈ G, there exists P ∈ P_{x} satisfying P ⊂ G, then G is open in X. Here, P_{x} is a weak base at x in X.

(2) P is an sn-network for X [15], if each member of P_{x} is a sequential neighborhood of x in X. Here, P_{x} is an sn-network at x in X.

(3) P is an so-network for X [15], if each member of P_{x} is sequentially open in X. Here, P_{x} is an so-network at x in X.

Definition 1.4. Let X be a space.

(1) X is a cosmic space [20] (resp., ℵ_{0}-space [20], sn-second countable space [9], so-second countable space, second countable space [5], ℵ-space [21], g-metrizable space [23]), if X has a countable network (resp., countable cs-network, countable sn-network, countable so-network, countable base, σ-locally finite cs-network, σlocally finite weak base).

(2) X is a sequential space [6], if each sequentially open subset of X is open.

Remark 1.1. [17] (1) For a space, weak base ⇒ sn-network ⇒ cs-network.

(2) An sn-network for a sequential space is a weak base.

Definition 1.5. Let f : X → Y be a mapping.

(1) f is an mssc-mapping [14], if X is a subspace of the product space ∏ _{n∈ℕ} X_{n} of a family {X_{n} : n ∈ ℕ} of metric spaces, and for each y ∈ Y, there exists a sequence \{V_{y,n} : n ∈ ℕ\} of open neighborhoods of y in Y such that each \overline{p_{n}(f^{−1} (V_{y,n}))} is a compact subset of X_{n}, where p_{n} : ∏ _{i∈ℕ} X_{i} → X_{n} is the projection.

(2) f is an 1-sequence-covering mapping [15] if, for each y ∈ Y, there exists x_{y} ∈ f^{−1} (y) such that whenever {y_{n} : n ∈ ℕ} is a sequence converging to y in Y there exists a sequence {x_{n} : n ∈ ℕ} converging to x_{y} in X with each x_{n} ∈ f^{−1} (y_{n}).

(3) f is a 2-sequence-covering mapping [15] if, for each y ∈ Y, x_{y} ∈ f^{−1} (y), and sequence {y_{n} : n ∈ ℕ} converging to y in Y, there exists a sequence {x_{n} : n ∈ ℕ} converging to x_{y} in X with each x_{n} ∈ f^{−1} (y_{n}).

(4) f is a sequence-covering mapping [22] if, for each convergent sequence S of Y, there exists a convergent sequence L of X such that f(L) = S. Note that a sequence-covering mapping is a strong sequence-covering mapping in the sense of [12].

(5) f is a pseudo-sequence-covering mapping [11] if, for each convergent sequence S of Y, there exists a compact subset K of X such that f(K) = S.

(6) f is a sequentially-quotient mapping [3] if, for each convergent sequence S of Y, there exists a convergent sequence L of X so that f(L) is a subsequence of S.

(7) f is a compact-covering mapping [20] if, for each compact subset K of Y, there exists a compact subset L of X such that f(L) = K.

(8) f is a π-mapping [2], if for each y ∈ Y and for each 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.

(9) f is a σ-mapping [18], if there exists a base B of X such that f(B) is a σ-locally finite family in Y.

Definition 1.6. [4] A space X is sequentially separable, if X has a countable subset D such that for each x ∈ X, there exists a sequence {x_{n} : n ∈ ℕ} in D converging to x. Here, the subset D is a sequentially dense subset of X.

For undefined terms, refer to [5] and [24].

2. Results

First, we characterize sequence-covering mssc-images of locally separable metric spaces by means of σ-locally finite cs-networks.

Theorem 2.1. The following are equivalent for a space X.

(1) X is a sequence-covering mssc-image of a locally separable metric space.

(2) X has a σ-locally finite cs-network consisting of cosmic spaces.

(3) X has a σ-locally finite cs-network consisting of ℵ_{0}-spaces.

Proof. (1)⇒(2). Let f : M → X be a sequence-covering mssc-mapping from a locally separable metric space M onto X, and {X_{n} : n ∈ ℕ} be the family of metric spaces satisfying that M is a subspace of ∏ _{n∈ℕ} X_{n}, and for each x ∈ X, there exists a sequence {V_{x,n} : n ∈ ℕ} of open neighborhoods of x in X such that each \overline{p_{n}(f^{−1} (V_{x,n} ))} is a compact subset of X_{n}, where p_{n} : ∏ _{i∈ℕ} X_{i} → X_{n} is the projection. Since M is locally separable metric, M = ⨁ _{λ∈Λ} M_{λ}, where each M_{λ} is a separable metric space by [5, 4.4.F]. Since each X_{n} is a metric space, X_{n} has a σ-locally finite base C_{n} = ⋃ {C_{n,i} : i ∈ ℕ}, where each C_{n,i} is locally finite. Assume, if necessary, that C_{n,i} ⊂ C_{n,i+1} for every i ∈ ℕ. For each n ∈ ℕ, set

B_{n} = { M ∩ ⋂_{i⩽n} p_{i}^{−1} (C_{i}) :

C_{i} ∈ ⋃_{j⩽n} C_{i,j} , i ⩽ n, M ∩ ⋂_{i⩽n} p_{i}^{−1} (C_{i}) ⊂ M_{λ} for some λ ∈ Λ} ,

set P_{n} = f(B_{n}), and set B = ⋃ {B_{n} : n ∈ ℕ}, P = ⋃ {P_{n} : n ∈ ℕ}. Then B is a base for M consisting of separable subsets. Assume, if necessary, that B is closed under finite intersections. We shall show that P is a σ-locally finite cs-network for X consisting of cosmic spaces by the following facts (a), (b), and (c).

(a) P is a cs-network for X.

Let S be a convergent sequence being eventually in U with U open in X. Since f is sequence-covering, there exists a convergent sequence L in M such that f(L) = S. Since L is eventually in B ⊂ f^{−1} (U) for some B ∈ B, S is eventually in f(B) ⊂ U. It implies that S is eventually in P ⊂ U with P = f(B) ∈ P. Therefore, P is a cs-network for X.

(b) P is σ-locally finite.

For each x ∈ X and n ∈ ℕ, set V_{x} = ⋂ _{i⩽n} V_{x,i} , then V_{x} is an open neighborhood of x in X. For each i ∈ ℕ, since \overline{p_{i}(f^{−1} (V_{x,i} ))} is a compact subset of X_{i} and C_{i,j} is locally finite, p_{i}(f^{−1} (V_{x,i} )) meets only finitely many members of C_{i,j} for every j ∈ ℕ. Then f^{−1} (V_{x,i} ) meets only finitely many members of {p_{i}^{−1} (C_{i}) : C_{i} ∈ ⋃ _{j⩽n} C_{i,j} }. Therefore, f^{−1} (V_{x}) meets only finitely many members of { ⋂ _{i⩽n} p_{i}^{−1} (C_{i}) : C_{i} ∈ ⋃ _{j⩽n} C_{i,j} , i ⩽ n}. It implies that f^{−1} (V_{x}) meets only finitely many members of B_{n}. Hence V_{x} meets only finitely many members of f(B_{n}), i.e., P_{n} is locally finite. It follows that P is σ-locally finite.

(c) Each P ∈ P is a cosmic space.

Set P = f(B) for some B ∈ B. Since B is separable, P is cosmic.

(2) ⇒(3). Let P = ⋃ {P_{n} : n ∈ ℕ} be a σ-locally finite cs-network for X consisting of cosmic spaces. Every locally finite family in a Lindelöf space is countable. Hence for each P ∈ P, {P ∩ P′: P′ ∈ P} is countable, and obviously it is a cs-network for P.

(3) ⇒(1). Let P = ⋃ {P_{n} : n ∈ ℕ} be a σ-locally finite cs-network for X consisting of ℵ_{0}-spaces, where each P_{n} = {P_{α_{n}} : α_{n} ∈ A_{n}} is a locally finite family. For each n ∈ ℕ, since each P_{α_{n}} is an ℵ_{0}-space, P_{α_{n}} has a countable cs-network P_{α_{n}} = {P_{α_{n,i}} : i ⩾ n}. For each i ⩾ n, set

Q_{α_{n,i}} = {P_{α_{n}} } ∪ {P_{α_{n,j}} : n ⩽ j ⩽ i} = {Q_{β} : β ∈ B_{α_{n,i}} },

where B_{α_{n,i}} is finite, and set

Q_{i} = {X} ∪ ( ⋃ {Q_{α_{j,i}} : α_{j} ∈ A_{j} , j ⩽ i}) = {Q_{β} : β ∈ B_{i}},

where B_{i} = {β_{0}} ∪ ( ⋃ {B_{α_{j,i}} : α_{j} ∈ A_{j} , j ⩽ i}) with Q_{β_{0}} = X. Since each P_{i} is locally finite and each Q_{α_{j,i}} is finite, Q_{i} is locally finite. Endow B_{i} with the discrete topology, then B_{i} is a metric space. Set

M = {b = (β_{i}) ∈ ∏_{i∈ℕ} B_{i} : there exists n ∈ ℕ and α_{n} ∈ A_{n} such that
Q_{β_{i}} = X if i < n, Q_{β_{i}} ∈ Q_{α_{n,i}} if i ⩾ n, and

{Q_{β_{i}} : i ⩾ n} forms a network at a point x_{b} in P_{α_{n}} }.

Then M, which is a subspace of the product space ∏ _{i∈ℕ} B_{i}, is a metric space. Since X is T_{1} and regular, x_{b} is unique for every b ∈ M. We define f : M → X by f(b) = x_{b} for every b ∈ M.

(a) f is onto.

For each x ∈ X, there exists n ∈ ℕ and α_{n} ∈ A_{n} such that x ∈ P_{α_{n}} . Since P_{α_{n}} is a countable cs-network for P_{α_{n}} , (P_{α_{n}} )_{x} = {Q_{β} ∈ P_{α_{n}} : x ∈ Q_{β}} is a countable network at x in P_{α_{n}} . We may assume that (P_{α_{n}} )_{x} = {P_{x,j} : j ∈ ℕ}, where P_{x,j} ∈ Q_{α_{n,i(j)}} with some i(j) ∈ ℕ satisfying i(j) < i(j + 1). For each i ∈ ℕ, take Q_{β_{i}} as follows.

(i) i < n: Q_{β_{i}} = X,
(ii) i ⩾ n: Q_{β_{i}} = P_{x,j} if i = i(j) for some j ∈ ℕ, and otherwise, Q_{β_{i}} = P_{α_{n}} .

Then {Q_{β_{i}} : i ⩾ n} − {P_{α_{n}} } = (P_{α_{n}} )_{x} − {P_{α_{n}} }. Therefore, {Q_{β_{i}} : i ⩾ n} forms a network at x in P_{α_{n}} . It implies that b = (β_{i}) ∈ M satisfying x = f(b), i.e., f is onto.

(b) f is continuous.

For each b = (β_{i}) ∈ M and x = f(b) ∈ U with U open in X. Then x = f(b) ∈ Q_{β_{k}} ⊂ U for some k ∈ ℕ. Set U_{b} = {c = (γ_{i}) ∈ M : γ_{k} = β_{k}}. Then U_{b} is open in M, and b ∈ U_{b}. For each c ∈ U_{b}, we find f(c) ∈ Q_{γ_{k}} = Q_{β_{k}} ⊂ U. It implies that f(U_{b}) ⊂ U, i.e., f is continuous.

(c) M is locally separable.

Let b = (β_{i}) ∈ M. Then there exists n ∈ ℕ and α_{n} ∈ A_{n} such that Q_{β_{i}} = X if i < n, Q_{β_{i}} ∈ Q_{α_{n,i}} if i ⩾ n, and {Q_{β_{i}} : i ⩾ n} forms a network at a point x_{b} in P_{α_{n}} . Set M_{b} = {c = (γ_{i}) ∈ M : γ_{n} = β_{n}}. Then M_{b} is open in M, and b ∈ M_{b}. For each c = (γ_{i}) ∈ M_{b}, there exists m ∈ ℕ and α_{m} ∈ A_{m} such that Q_{γ_{i}} = X if i < m, Q_{γ_{i}} ∈ Q_{α_{m,i}} if i ⩾ m, and {Q_{γ_{i}} : i ⩾ m} forms a network at a point x_{c} in P_{α_{m}} . It follows from Q_{γ_{n}} = Q_{β_{n}} that P_{α_{m}} ∩ P_{α_{n}} ≠ ∅. Since P_{α_{n}} is an ℵ_{0}-space and P_{m} is locally finite, C_{m} = {α_{m} ∈ A_{m} : P_{α_{m}} ∩ P_{α_{n}} ≠ ∅} is countable for every m ∈ ℕ. Then E_{i} = {β_{0}} ∪ ( ⋃ {B_{α_{j,i}} : α_{j} ∈ C_{j} , j ⩽ i}) is countable. It implies that {β_{1}} × ⋯× {β_{n−1} } × ∏ _{i⩾n} E_{i} is hereditarily separable. Since M_{b} ⊂ {β_{1}}×⋯ ×{β_{n−1} }×∏ _{i⩾n} E_{i}, M_{b} is separable. Therefore, M is locally separable.

(d) f is an mssc-mapping.

For each x ∈ X and each i ∈ ℕ, since P_{i} is locally finite, there exists an open neighborhood V_{x,i} of x in X such that D_{i} = {α_{i} ∈ A_{i} : P_{α_{i}} ∩ V_{x,i} ≠ ∅} is finite. Then F_{i} = {β_{0}} ∪ ( ⋃ {B_{α_{j,i}} : α_{j} ∈ D_{j} , j ⩽ i}) is finite. Since p_{i}(f^{−1} (V_{x,i} )) ⊂ F_{i}, \overline{p_{i}(f^{−1} (V_{x,i} ))} is compact. It implies that f is an mssc-mapping.

(e) f is sequence-covering.

For each convergent sequence S in X, since P is a σ-locally finite cs-network for X, there exists n ∈ ℕ and α_{n} ∈ A_{n} such that S is eventually in P_{α_{n}} ∈ P_{n}. Then L_{α_{n}} = S ∩ P_{α_{n}} is a convergent sequence in P_{α_{n}} . For each i ⩾ n, we find that ⋃ {Q_{α_{n,i}} : i ⩾ n} is a σ-locally finite cs-network for P_{α_{n}} satisfying P_{α_{n}} ∈ Q_{α_{n,i}} ⊂ Q_{α_{n,i+1}} . It follows from the proof (3)⇒(2) of [13, Theorem 5.1] that there exists a convergent sequence H_{α_{n}} in M_{α_{n}} such that f_{α_{n}} (H_{α_{n}} ) = L_{α_{n}} , where

M_{α_{n}} = {c = (γ_{i})_{i⩾n} ∈ ∏_{i⩾n} B_{α_{n,i}} : {Q_{γ_{i}} : i ⩾ n} forms a network
at a point x_{c} in P_{α_{n}} },

and f_{α_{n}} : M_{α_{n}} → P_{α_{n}} defined by f_{α_{n}} (c) = x_{c} for every c ∈ M_{α_{n}} . For each c = (γ_{i})_{i⩾n} ∈ H_{α_{n}} , set b_{c} = (β_{i})_{i∈ℕ} , where Q_{β_{i}} = X if i < n and β_{i} = γ_{i} if i ⩾ n, and set H = {b_{c} : c ∈ H_{α_{n}} }. Then H is a convergent sequence in M and f(H) = L_{α_{n}} . Since S is eventually in P_{α_{n}} , S − P_{α_{n}} is finite. Then S − P_{α_{n}} = f(F) with some finite subset F of M. Set L = H ∪ F, then L is a convergent sequence in M satisfying f(L) = S. It implies that f is sequence-covering. □

Remark 2.1. The argument for cs-networks in the proof(2) ⇒(3) of Theorem 2.1 can not apply to cs^{∗}-networks or cfp-networks.

Corollary 2.1. The following are equivalent for a space X.

(1) X is a sequence-covering, quotient mssc-image of a locally separable metric space.

(2) X is a sequential space having a σ-locally finite cs-network consisting of cosmic spaces.

(3) X is a sequential space having a σ-locally finite cs-network consisting of ℵ_{0}-spaces.

Proof. (1) ⇒ (2). Since X is a quotient image of a locally separable metric space, X is a sequential space by [6, Proposition 1.2]. Then X is a sequential space having a σ-locally finite cs-network consisting of cosmic spaces by Theorem 2.1.

(2) ⇒ (3). As in the proof(2) ⇒(3) of Theorem 2.1.

(3) ⇒(1). It follows from Theorem 2.1 that X is a sequence-covering msscimage of a locally separable metric space under some mapping f. Since f is a sequence-covering mapping onto a sequential space, f is a quotient mapping by [17, Lemma 3.5]. It implies that X is a sequence-covering, quotient mssc-image of a locally separable metric space. □

Next, we characterize 1-sequence-covering mssc-images of locally separable metric spaces by means of σ-locally finite sn-networks.

Theorem 2.2. The following are equivalent for a space X.

(1) X is an 1-sequence-covering mssc-image of a locally separable metric space.

(2) X has a σ-locally finite sn-network consisting of cosmic spaces.

(3) X has a σ-locally finite sn-network consisting of sn-second countable spaces.

Proof. (1) ⇒ (2). Let f : M → X be an 1-sequence-covering mssc-mapping from a locally separable metric space M onto X. For each x ∈ X, let a_{x} ∈ f^{−1} (x) satisfying that whenever {x_{n} : n ∈ ℕ} is a sequence converging to x in X there exists a sequence {a_{n} : n ∈ ℕ} converging to a_{x} in M with each a_{n} ∈ f^{−1} (x_{n}). By using notations in the proof (1) ⇒ (2) of Theorem 2.1 again, let Q_{x} = {P ∈ P : P = f(B) with a_{x} ∈ B ∈ B}, and let Q = ⋃ {Q_{x} : x ∈ X}. We shall prove that Q is a σ-locally finite sn-network for X consisting of cosmic spaces by the following facts (a), (b), (c) for every x ∈ X, and (d), (e).

(a) Q_{x} is a network at x in X.

It is clear that x ∈ ⋂ Q_{x}. Let x ∈ U with U open in X, then a_{x} ∈ f^{−1} (U). Since B is a base for M, a_{x} ∈ B ⊂ f^{−1} (U) for some B ∈ B. Set Q = f(B), then Q ∈ Q_{x} and x ∈ Q ⊂ U. It implies that Q_{x} is a network at x in X.

(b) If Q_{1}, Q_{2} ∈ Q_{x}, then Q ⊂ Q_{1} ∩ Q_{2} for some Q ∈ Q_{x}.

Set Q_{1} = f(B_{1}), Q_{2} = f(B_{2}), where B_{1}, B_{2} ∈ B with a_{x} ∈ B_{1} and a_{x} ∈ B_{2}.

Since B is a base for M, a_{x} ∈ B ⊂ B_{1} ∩ B_{2} for some B ∈ B. Set Q = f(B), then Q ∈ Q_{x} and Q ⊂ Q_{1} ∩ Q_{2}.

(c) Each Q ∈ Q_{x} is a sequential neighborhood of x.

Set Q = f(B) with a_{x} ∈ B ∈ B. For each convergent sequence S converging to x, there exists a convergent sequence L converging to a_{x} in M such that f(L) = S. Since L is eventually in B, S is eventually in Q. It implies that Q is a sequential neighborhood of x.

(d) Q is σ-locally finite.

Since Q ⊂ P and P is σ-locally finite, Q is σ-locally finite.

(e) Each Q ∈ Q is a cosmic space.

Set Q = f(B) for some B ∈ B. Since B is separable, Q is cosmic.

(2) ⇒ (3). As in the proof(2) ⇒ (3) of Theorem 2.1.

(3) ⇒ (1). Let P = ⋃ {P_{n} : n ∈ ℕ} be a σ-locally finite sn-network for X consisting of ℵ_{0}-spaces. By using notations and arguments in the proof (3) ⇒ (1) of Theorem 2.1 again, since each sn-network is also a cs-network, it suffices to prove that the mapping f is 1-sequence-covering.

For each x ∈ X, since P is a σ-locally finite sn-network for X, there exists n ∈ ℕ and α_{n} ∈ A_{n} such that P_{α_{n}} is a sequential neighborhood of x. Then ⋃ {Q_{α_{n,i}} : i ⩾ n} is a σ-locally finite sn-network for P_{α_{n}} . It implies that f_{α_{n}} is 1-sequence-covering by [13, Theorem 2.1]. Hence, there exists c_{x} = (γ_{x,i} )_{i⩾n} ∈ f_{α_{n}}^{−1} (x) such that whenever {x_{m} : m ∈ ℕ} is a sequence converging to x in P_{α_{n}} there exists a sequence {c_{m} : m ∈ ℕ} converging to c_{x} in M_{α_{n}} with each c_{m} ∈ f_{α_{n}}^{−1} (x_{m}). Set b_{x} = (β_{x,i} ), where Q_{β_{x,i}} = X if i < n and β_{x,i} = γ_{x,i} if i ⩾ n, then b_{x} ∈ f^{−1} (x). Let {y_{m} : m ∈ ℕ} be a sequence in X converging to x. Since P_{α_{n}} is a sequential neighborhood of x, there exists m_{0} ∈ ℕ such that {y_{m} : m ⩾ m_{0}} ⊂ P_{α_{n}} is a sequence converging to x in P_{α_{n}} . Then there exists a sequence {c_{m} : m ⩾ m_{0}} in M_{α_{n}} converging to c_{x} and c_{m} ∈ f_{α_{n}}^{−1} (y_{m}) for each m ⩾ m_{0}. For each c_{m} = (γ_{m,i} )_{i⩾n}, set b_{m} = (β_{m,i} ), where Q_{β_{m,i}} = X if i < n and β_{m,i} = γ_{m,i} if i ⩾ n. Then b_{m} ∈ M and f(b_{m}) = y_{m} for each m ⩾ m_{0}. For each m < m_{0}, take some b_{m} ∈ f^{−1} (y_{m}). Then {b_{m} : m ∈ ℕ} is a sequence in M converging to b_{x} and b_{m} ∈ f^{−1} (y_{m}) for each m ∈ ℕ. It implies that f is 1-sequence-covering. □

Corollary 2.2. The following are equivalent for a space X.

(1) X is an 1-sequence-covering, quotient mssc-image of a locally separable metric space.

(2) X has a σ-locally finite weak base consisting of cosmic spaces.

(3) X has a σ-locally finite weak base consisting of sn-second countable spaces.

Proof. (1) ⇒ (2). Since X is a quotient image of a locally separable metric space, X is a sequential space by [6, Proposition 1.2]. Then X is a sequential space having a σ-locally finite sn-network P consisting of cosmic spaces by Theorem 2.2. It follows from Remark 1.1 that P is a weak base for X. Therefore, X has a σ-locally finite weak base consisting of cosmic spaces.

(2) ⇒ (3). Since X has a σ-locally weak base, X is a sequential space. It follows from Theorem 2.2 that X is a sequential space having a σ-locally finite sn-network P consisting of sn-second countable spaces. By Remark 1.1, P is a weak base for X. It implies that X has a σ-locally finite weak base consisting of sn-second countable spaces.

(3) ⇒ (1). It follows from Theorem 2.2 that X is an 1-sequence-covering msscimage of a locally separable metric space under some mapping f. Since X has a σ-locally finite weak base, X is a sequential space. Then f is an 1-sequence-covering mapping onto a sequential space, and so f is a quotient mapping by [17, Lemma 3.5]. It implies that X is an 1-sequence-covering, quotient mssc-image of a locally separable metric space. □

Remark 2.2. We can replace “cosmic spaces” in Theorem 2.2 and Corollary 2.2 by “ℵ_{0}-spaces”.

In the following, we characterize 2-sequence-covering mssc-images of locally separable metric spaces by means of σ-locally finite so-networks.

Theorem 2.3. The following are equivalent for a space X.

(1) X is a 2-sequence-covering mssc-image of a locally separable metric space.

(2) X has a σ-locally finite so-network consisting of cosmic spaces.

(3) X has a σ-locally finite so-network consisting of so-second countable spaces.

Proof. (1) ⇒ (2). Let f : M → X be a 2-sequence-covering mssc-mapping from a locally separable metric space M onto X. For each x ∈ X, by using notations in the proof(1) ⇒ (2) of Theorem 2.1 again, let B_{x} = {B ∈ B : f^{−1} (x) ∩ B ≠ ∅}, and let R_{x} be the family of all finite intersections of members of f(B_{x}). We shall prove that R = ⋃ {R_{x} : x ∈ X} is a σ-locally finite so-network for X consisting of cosmic spaces by the following facts (a), (b), (c) for every x ∈ X and (d), (e).

(a) R_{x} is a network at x in X.

This is obvious because B_{x} is a base for f^{−1} (x).

(b) If R_{1}, R_{2} ∈ R_{x}, then R ⊂ R_{1} ∩ R_{2} for some R ∈ R_{x}.

This is obvious by choosing R = R_{1} ∩ R_{2}.

(c) Each R ∈ R_{x} is sequentially open.

Let B ∈ B_{x}, y ∈ f(B), and S be a convergent sequence converging to y. Since y ∈ f(B), f^{−1} (y) ∩ B ≠ ∅. Take some a_{y} ∈ f^{−1} (y) ∩ B. Then there exists a convergent sequence L converging to a_{y} in M such that f(L) = S. Since L is eventually in B, S is eventually in f(B). It implies that f(B) is sequentially open, i.e., every member of f(B_{x}) is sequentially open. Because R is some finite intersection of members of f(B_{x}), we find that R is sequentially open.

(d) R is σ-locally finite.

Since ⋃ {f(B_{x}) : x ∈ X} ⊂ P and P is σ-locally finite, ⋃ {f(B_{x}) : x ∈ X} is σ-locally finite. It implies that R is σ-locally finite.

(e) Each R ∈ R is a cosmic space.

For each B ∈ B_{x}, since B is separable, f(B) is cosmic, i.e., every member of f(B_{x}) is cosmic. It implies that R is cosmic.

(2) ⇒ (3). As in the proof(2) ⇒(3) of Theorem 2.1.

(3) ⇒ (1). Let P = ⋃ {P_{n} : n ∈ ℕ} be a σ-locally finite so-network for X consisting of ℵ_{0}-spaces. By using notations and arguments in the proof (3) ⇒ (1) of Theorem 2.1 again, since each so-network is also a cs-network, it suffices to prove that the mapping f is 2-sequence-covering.

For each x ∈ X and each b_{x} ∈ f^{−1} (x), let b_{x} = (β_{x,i} ). Then there exists some n ∈ ℕ and α_{n} ∈ A_{n} such that Q_{β_{x,i}} = X if i < n, Q_{β_{x,i}} ∈ Q_{α_{n,i}} if i ⩾ n, and {Q_{β_{x,i}} : i ⩾ n} forms a network at x in P_{α_{n}} . Set c_{x} = (β_{x,i} )_{i⩾n}, then c_{x} ∈ f_{α_{n}}^{−1} (x). Since {Q_{α_{n,i}} : i ⩾ n} is a σ-locally finite so-network for P_{α_{n}} , f_{α_{n}} is a 2-sequence-covering by [13, Theorem 3.1]. Let {x_{m} : m ∈ ℕ} be a sequence converging to x in X. Since P_{α_{n}} is sequentially open, there exists m_{0} ∈ ℕ such that {x_{m} : m ⩾ m_{0}} is a sequence converging to x in P_{α_{n}} . Then there exists a sequence {c_{m} : m ⩾ m_{0}} in M_{α_{n}} converging to c_{x} and c_{m} ∈ f_{α_{n}}^{−1} (x_{m}) for each m ⩾ m_{0}. For each c_{m} = (γ_{m,i} )_{i⩾n}, set b_{m} = (β_{m,i} ), where Q_{β_{m,i}} = X if i < n, and β_{m,i} = γ_{m,i} if i ⩾ n. Then b_{m} ∈ M and f(b_{m}) = x_{m} for each m ⩾ m_{0}. For each m < m_{0}, take some b_{m} ∈ f^{−1} (x_{m}). Then {b_{m} : m ∈ ℕ} is a sequence in M converging to b_{x} and b_{m} ∈ f^{−1} (x_{m}) for each m ∈ ℕ. It implies that f is 2-sequence-covering. □

Corollary 2.3. The following are equivalent for a space X.

(1) X is a 2-sequence-covering, quotient mssc-image of a locally separable metric space.

(2) X has a σ-locally finite base consisting of cosmic spaces.

(3) X has a σ-locally finite base consisting of second countable spaces.

Proof. (1) ⇒ (2). Since X is a quotient image of a locally separable metric space, X is a sequential space by [6, Proposition 1.2]. It follows from Theorem 2.3 that X is a sequential space having a σ-locally finite so-network P consisting of cosmic spaces. For each P ∈ P, since X is sequential and P is sequential open, P is open in X. Hence P is a σ-locally finite base for X consisting of cosmic spaces.

(2) ⇒(3). It follows from Theorem 2.3 that X has a σ-locally finite so-network P consisting of so-second countable spaces. Since X has a σ-locally finite base, X is sequential. It implies that every P ∈ P is open. Then P is a σ-locally finite base consisting of so-second countable spaces.

Let P ∈ P and Q be a countable so-network for P. Since P is open, P is a sequential space by [6, Proposition 1.9]. Then every Q ∈ Q is open in P. Hence Q is a countable base for P. It implies that P is a second countable space.

By the above, X has a σ-locally finite base consisting of second countable spaces.

(3) ⇒ (1). It follows from Theorem 2.3 that X is a 2-sequence-covering msscimage of a locally separable metric space under some mapping f. Since X has a σ-locally finite base, X is sequential. Then f is a 2-sequence-covering mapping onto a sequential space, and so f is a quotient mapping by [17, Lemma 3.5]. It implies that X is a 2-sequence-covering, quotient mssc-image of a locally separable metric space. □

Remark 2.3. We can replace “cosmic spaces” in Theorem 2.3 and Corollary 2.3 by “ℵ_{0}-spaces”, or “sn-second countable spaces”.

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Mathematics Faculty (Received 11 10 2008) Dongthap University Caolanh City Dongthap Province Vietnam

    nguyendungtc@yahoo.com
    nvdung@staff.dthu.edu.vn