PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 93 (107) (2013), 133–144 DOI: 10.2298/PIM1307133A

L-PONOMAREV SYSTEM AND IMAGES OF LOCALLY SEPARABLE METRIC SPACES

Tran Van An and Luong Quoc Tuyen

Communicated by Miloš Kurilić

Abstract. We introduce the notion of an L-Ponomarev system (f, M, X, P_{n}^{∗}), and give characterizations of certain msss-images (resp., mssc-images) of locally separable metric spaces. As an application, we get a new characterization of quotient msss-images (mssc-images) of locally separable metric spaces, which is helpful in solving Velichko’s question (1987).

1. Introduction

Lin in [15] introduced the concept of msss-maps (resp., mssc-maps) to characterize spaces with certain σ-locally countable (resp., σ-locally finite) networks by msss-images (resp., mssc-images) of metric spaces. After that, some characterizations for certain msss-images (resp., mssc-images) of metric (or semi-metric) spaces are obtained by many authors ([10, 13, 14], for example).

Velichko [26] proved that a space X is a pseudo-open s-image of a locally separable metric space iff X is a locally separable space which is a pseudo-open simage of a metric space, and posed the following interesting question about quotient and s-images of metric spaces.

Question 1.1. Find a Φ-property such that a space X is a quotient and simage of a metric and Φ-space iff X is a Φ-space which is a quotient and s-image of a metric space.

Recently, Dung gave some characterizations for certain msss-images (resp., mssc-images) of locally separable metric spaces in the class of regular and T_{1}-spaces (see in [3, 4]). This leads us to consider the following question.

2010 Mathematics Subject Classification: 54C10, 54D55, 54E40, 54E99.

Key words and phrases: so-network, sn-network, cs-network, cfp-network, cs^{∗}-network, 2-sequence-covering, 1-sequence-covering, sequence-covering, compact-covering, sequentiallyquotient, msss-map, mssc-map.

Question 1.2. Find a Φ-property such that a space X is a quotient and msssimage (mssc-image) of a metric and Φ-space iff X is a Φ-space which is a quotient and msss-image (resp., mssc-image) of a metric space.

In this paper, we introduce the notion of a generalized Ponomarev system (f, M, X, P_{n}^{∗}), calling it an L-Ponomarev system, and then prove some statements concerning the properties of such systems corresponding to σ-locally finite and σ-locally countable Lindelöf networks. As an application, we get a new characterization of quotient msss-images (mssc-images) of locally separable metric spaces, give an affirmative answer to Question 1.2, and we get an affirmative answer to Question 2.17 from [3].

Throughout this paper, all spaces are assumed to be Hausdorff, all maps are continuous and onto, ℕ denotes the set of all natural numbers. Let K ⊂ X and P be a collection of subsets of X, we denote (P)_{x} = {P ∈ P : x ∈ P }, P_{K} = {P ∈ P : P ∩ K ≠ ∅}. For a sequence {x_{n}} converging to x and P ⊂ X, we say that {x_{n}} is eventually in P if {x} ∪ {x_{n} : n ⩾ m} ⊂ P for some m ∈ ℕ, and {x_{n}} is frequently in P if some subsequence of {x_{n}} is eventually in P.

Definition 1.1. [2, 17] Let P = ⋃ {P_{x} : x ∈ X} be a cover of a space X. Assume that P satisfies the following (a) and (b) for every x ∈ X.

(a) P_{x} is a network at x.

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

(1) P is a weak base for X, if for G ⊂ X, G is open in X iff for every x ∈ G, there exists P ∈ P_{x} such that P ⊂ G.

(2) P is an sn-network (resp., so-network) for X, if every element of P_{x} is a sequential neighborhood of x (resp., sequentially open in X) for every x ∈ X.

Definition 1.2. Let X be a space and P be a cover of X.

(1) P is a Lindelöf (resp., compact) cover, if each element of P is Lindelöf (resp., compact).

(2) X is an ℵ_{0}-space, if X is a regular space with a countable cs^{∗}-network.

(3) X is an H-ℵ_{0}-space, if X has a countable cs^{∗}-network.

Definition 1.3. Let f : X → Y be a map.

(1) f is weak-open [27], if there exists a weak base B = ⋃ {B_{y} : y ∈ Y } for Y, and for every y ∈ Y, there exists x ∈ f^{−1} (y) such that for each open neighborhood U of x, B ⊂ f(U) for some B ∈ B_{y}.

(2) f is 1-sequence-covering [17], if for each y ∈ Y, there is x ∈ f^{−1} (y) such that each sequence converging to y is an image of some sequence converging to x.

(3) f is 2-sequence-covering [17], if for every y ∈ Y, x_{y} ∈ f^{−1} (y), and sequence {y_{n}} converging to y in Y, there exists a sequence {x_{n}} converging to x_{y} in X with each x_{n} ∈ f^{−1} (y_{n}).

L-PONOMAREV SYSTEM AND IMAGES OF LOCALLY SEPARABLE METRIC SPACES 135

(4) f is an msss-map (resp., mssc-map) [15], if X is a subspace of the product space ∏ _{i∈ℕ} X_{i} of a family {X_{i} : i ∈ ℕ} of metric spaces and for each y ∈ Y, there is a sequence {V_{i} : i ∈ ℕ} of open neighborhood’s of y such that each p_{i}f^{−1} (V_{i}) is separable in X_{i} (resp., each cl(p_{i}f^{−1} (V_{i})) is compact in X_{i}).

Definition 1.4. For a cover P of a space X, let (P) be a (certain) coveringproperty of P. Let us say that P has property σ-(P), if P can be expressed as ⋃ {P_{n} : n ∈ ℕ}, where each P_{n} having the property (P) and P_{n} ⊂ P_{n+1} for all n ∈ ℕ.

For some undefined or related concepts, we refer the reader to [18].

2. Main results

From now on, let us restrict the properties (P) and α(P) to the following.

(1) (P) are locally finite, locally countable.

(2) α(P) is mssc if (P) is locally finite, and α(P) is msss if (P) is locally countable.

Notation 2.1. Let P = ⋃ {P_{n} : n ∈ ℕ} be a Lindelöf network having property σ-(P) for a space X. For each n ∈ ℕ, we put P_{n}^{∗} = {X} ∪ P_{n} = {P_{α} : α ∈ Λ_{n}} and endow Λ_{n} with the discrete topology. Assume that for each x ∈ X, there exists a network {P_{α_{n}} : n ∈ ℕ} at x with α_{n} ∈ Λ_{n}. Then,

M = {α = (α_{n}) ∈ ∏ _{n∈ℕ} Λ_{n} : {P_{α_{n}} } forms a network at some point x_{α} ∈ X}

is a metric space and the point x_{α} is unique in X for every α ∈ M. Define f : M → X by f(α) = x_{α}. Let us call (f, M, X, P_{n}^{∗}) an L-Ponomarev system.

Remark 2.1. (1) Let P = ⋃ {P_{n} : n ∈ ℕ} be a Lindelöf network of X, where each P_{n} having property (P). Then, P is a Lindelöf network has property σ-(P).

(2) If (f, M, X, P_{n}^{∗}) an L-Ponomarev system, then f is an s-map.

Lemma 2.1. If P is a cs-network having property σ-(P), then P is a cfpnetwork.

Proof. Let P = ⋃ {P_{n} : n ∈ ℕ} be a cs-network having property σ-(P) for X, and K ⊂ V with K is compact and V is open in X. Since P is a cs-network having property σ-(P), K has a countable cs-network. Thus, K is metrizable. By [19, Lemma 1.2], for each x ∈ K, there exists P_{x} ∈ P such that x ∈ int_{K} (P_{x} ∩ K) ⊂ P_{x} ⊂ V. By the regularity of K, for each x ∈ K, there exists an open neighborhood V_{x} in K such that x ∈ V_{x} ⊂ cl_{K} (V_{x}) ⊂ int_{K} (P_{x} ∩ K). Since K is compact, there exists a finite subset F of K such that K ⊂ ⋃ _{x∈F} V_{x}. Thus, {P_{x} : x ∈ F } is a cfp-cover of K and ⋃ _{x∈F} P_{x} ⊂ U. Therefore, P is a cfp-network. □

Lemma 2.2. If X has a Lindelöf cs^{∗}-network with property σ-(P), then X has a Lindelöf cs-network with property σ-(P).

Proof. Let P = ⋃ {P_{i} : i ∈ ℕ} be a Lindelöf cs^{∗}-network having property σ-(P) for X. Since each element of P_{i} is Lindelöf, each P_{i} is star-countable. It follows from [22, Lemma 2.1] that for each i ∈ ℕ, P_{i} = ⋃ {Q_{α}^{(i)} : α ∈ Λ_{i}}, where Q_{α}^{(i)} is a countable subfamily of P_{i} for all α ∈ Λ_{i} and ( ⋃ Q_{α}^{(i)} ) ∩ ( ⋃ Q_{β}^{(i)} ) = ∅ for all α ≠ β. For each i ∈ ℕ and α ∈ Λ_{i}, we put

R_{α}^{(i)} = {⋃ F : F is a finite subfamily of Q_{α}^{(i)} } .

Since each R_{α}^{(i)} is countable, we can write R_{α}^{(i)} = {R_{α,j}^{(i)} : j ∈ ℕ}. Now, for each i, j ∈ ℕ, put F_{j}^{(i)} = {R_{α,j}^{(i)} : α ∈ Λ_{i}}, and denote G = ⋃ {F_{j}^{(i)} : i, j ∈ ℕ}. Then, each R_{α,j}^{(i)} is Lindelöf and each family F_{j}^{(i)} has property (P). Now, we shall show that G is a cs-network. In fact, let {x_{n}} be a sequence converging to x ∈ U with U is open in X. Since P is a point-countable cs^{∗}-network, it follows from [25, Lemma 3] that there exists a finite family A ⊂ (P)_{x} such that {x_{n}} is eventually in ⋃ A ⊂ U. Furthermore, since A is finite and P_{i} ⊂ P_{i+1} for all i ∈ ℕ, there exists i ∈ ℕ such that A ⊂ P_{i}. So, there exists unique α ∈ Λ_{i} such that A ⊂ Q_{α}^{(i)} , and ⋃ A ∈ R_{α}^{(i)} . Thus, ⋃ A = R_{α,j}^{(i)} for some j ∈ ℕ. Hence, ⋃ A ∈ G, and G is a cs-network. It follows from Remark 2.1(1) G is a Lindelöf cs-network having property σ-(P). □

Lemma 2.3. Let f : M → X be a α(P)-map, and M be a locally separable metric space. Then,

(1) X has a Lindelöf cs^{∗}-network with property σ-(P), if f is sequentiallyquotient.

(2) X has a Lindelöf sn-network with property σ-(P), if f is 1-sequence-covering.

(3) X has a Lindelöf so-network with property σ-(P), if f is 2-sequence-covering.

Proof. By [15, Lemma 1.2] and by the proof of (3) ⇒ (1) in [12, Theorem 4], there exists a base B of M such that F = f(B) is a network for X, and F can be expressed as ⋃ {F_{n} : n ∈ ℕ}, where each F_{n} has property (P). Since M is locally separable, for each a ∈ M, there exists a separable open neighborhood U_{a}. Denote

C = {B ∈ B : B ⊂ U_{a}, a ∈ M}.

Then, C ⊂ B and C is a separable base for M. If put P = f(C), then P ⊂ F, and it follows from Remark 2.1(1) that P is a Lindelöf network having property σ-(P). Thus, P can be expressed as ⋃ {P_{n} : n ∈ ℕ}, where each P_{n} having the property (P) and P_{n} ⊂ P_{n+1} for all n ∈ ℕ. Furthermore, we have

(1) If f is sequentially-quotient, then since C is a base for M, P is a cs^{∗}-network. Therefore, X has a Lindelöf cs^{∗}-network with property σ-(P).

(2) If f is 1-sequence-covering, then for each x ∈ X, there exists a_{x} ∈ f^{−1} (x) such that each sequence converging to x is an image of a sequence converging to a_{x}. Now, for each x ∈ X, we put G_{x} = {f(B) : a_{x} ∈ B ∈ C}, G = ⋃ {G_{x} : x ∈ X}. Then, G ⊂ P and G is an sn-network. For each n ∈ ℕ, we put G_{n} = G ∩ P_{n}. Then, ⋃ {G_{n} : n ∈ ℕ} is a Lindelöf sn-network having property σ-(P) for X.

(3) If f is 2-sequence-covering, then for each x ∈ X, we put

C_{x} = {B ∈ C : B ∩ f^{−1} (x) ≠ ∅},

and let G_{x} be the family of all finite intersections of members of f(C_{x}), and G = ⋃ {G_{x} : x ∈ X}. Then, G ⊂ P and G is an so-network. For each n ∈ ℕ, we put G_{n} = G ∩ P_{n}. Then, ⋃ {G_{n} : n ∈ ℕ} is a Lindelöf so-network having property σ-(P) for X. □

Lemma 2.4. Let P = ⋃ {P_{n} : n ∈ ℕ} be a Lindelöf network having property σ-(P) and (f, M, X, P_{n}^{∗}) be an L-Ponomarev system. Then, the following statements hold.

(1) f is a α(P)-map.

(2) M is locally separable.

(3) f is sequence-covering compact-covering, if P is a cs-network.

(4) f is 1-sequence-covering compact-covering, if P is an sn-network.

(5) f is 2-sequence-covering compact-covering, if P is an so-network.

Proof. (1) Similar to the proof of [12, Theorem 4] and [14, Theorem 2.1].

(2) Let a = (α_{i}) ∈ M. Then, {P_{α_{i}} } is a network at some point x_{a} ∈ X. Thus, there exists i_{0} ∈ ℕ such that P_{α_{i_{0}}} is Lindelöf. Put

U_{a} = M ∩ {(β_{i}) ∈ ∏ _{i∈ℕ} Λ_{i} : β_{i} = α_{i}, i ⩽ i_{0}}.

Then, U_{a} is an open neighborhood of a in M. Now, for each i ⩽ i_{0}, put Δ_{i} = {α_{i}}, and for each i > i_{0}, we put Δ_{i} = {α ∈ Λ_{i} : P_{α} ∩ P_{α_{i_{0}}} ≠ ∅}. Then, U_{a} ⊂ ∏ _{i∈ℕ} Δ_{i}. Furthermore, since each P_{i} having property (P) and P_{α_{i_{0}}} is Lindelöf, Δ_{i} is countable for every i > i_{0}. Thus, U_{a} is separable, and M is locally separable.

(3) Let P be a cs-network. Then,

(3.1) f is sequence-covering. Let S = {x_{n} : n ∈ ℕ} be a sequence converging to x in X. Since P is a point-countable cs-network, we can write

{P ∈ P : S is eventually in P } = {P_{i} : i ∈ ℕ}.

On the other hand, since P_{i} ⊂ P_{i+1} for all i ∈ ℕ, we can choose sequence {i_{n}} ⊂ ℕ such that i_{n} < i_{n+1} , and P_{n} ∈ P_{i_{n}} for every n ∈ ℕ. Now, for each j ∈ ℕ, we take

F_{α_{j}} = \{ \begin{matrix}P_{n}, & if j = i_{n}, \\ X, & if j ≠ i_{n}, \end{matrix}

and a = (α_{i}) ∈ ∏ _{i∈ℕ} Λ_{i}. Then f(a) = x and S is eventually in each F_{α_{i}} . Now, for each n ∈ ℕ, put B_{n} = {(γ_{i}) ∈ M : γ_{i} = α_{i} for each i ⩽ n}. It is easy to check that {B_{n}} is a decreasing neighborhood base at a in M and f(B_{n}) = ⋂ _{i⩽n} P_{α_{i}} for all n ∈ ℕ. Because S is eventually in each f(B_{n}), it follows from [8, Lemma 6] that for each n ∈ ℕ, there exists a_{n} ∈ f^{−1} (x_{n}) such that the sequence {a_{n}} converging to a in M. Therefore, f is sequence-covering.

(3.2) f is compact-covering. Let K be a compact subset of X. Since P is a Lindelöf cs-network having property σ-(P), it follows from Lemma 2.1 that P is a cfp-network for X. Furthermore, since P_{K} is countable, we can put

{Q ⊂ P_{K} : Q is a finite cfp-cover of K} = {Q_{i} : i ∈ ℕ}.

Since Q_{n} ⊂ P and P_{n} ⊂ P_{n+1} for all n ∈ ℕ, then we can choose a sequence {i_{n}} ⊂ ℕ such that i_{n} < i_{n+1} , and Q_{n} ⊂ P_{i_{n}} for every n ∈ ℕ. Now, we choose a sequence {A_{i}} as follows

A_{j} = \{ \begin{matrix}Q_{n}, & if j = i_{n}, \\ \{X\}, & if j ≠ i_{n}. \end{matrix}

Since each A_{i} is a cfp-cover for K, there exists a finite subfamily H_{i} = {P_{α} }_{α∈Γ_{i}} of A_{i} and a cover {F_{α} }_{α∈Γ_{i}} of K consisting of closed subset of K satisfying that each F_{α} ⊂ P_{α}. Put L = {a = (α_{i}) ∈ ∏ _{i∈ℕ} Γ_{i} : ⋂ _{i∈ℕ} F_{α_{i}} ≠ ∅}. Then, we have

(3.2.1) L ⊂ M, and f(L) ⊂ K. Suppose a = (α_{i}) ∈ L, then ⋂ _{i∈ℕ} F_{α_{i}} ≠ ∅. Pick x_{a} ∈ ⋂ _{i∈ℕ} F_{α_{i}} . Now we will show that {P_{α_{i}} } is a network at x_{a} in X. Then, a ∈ M and f(a) = x_{a} ∈ K, so L ⊂ M and f(L) ⊂ K. Indeed, let V be a neighborhood of x_{a} in X. Since K is a regular subspace of X, there exists an open neighborhood W of x_{a} in K such that cl_{K} (W) ⊂ V. Since cl_{K} (W) is a compact subset of K, there exists a finite collection Q′ of P_{K} such that Q′ is a cfp-cover of cl_{K} (W) and ⋃ Q′ ⊂ V. On the other hand, since K − W is a compact subset of K satisfying K − W ⊂ X − {x_{a}}, there exists a finite collection Q′′ of P_{K} such that Q′′ is a cfp-cover for K − W and ⋃ Q′′ ⊂ X − {x_{a}}. Put Q = Q′ ∪ Q′′. Then, Q is a cfp-cover for K, and so Q = Q_{k} for some k ∈ ℕ. But x_{a} ∈ F_{α_{k}} ⊂ P_{α_{k}} ∈ Q_{k}, thus P_{α_{k}} ∈ Q′ and P_{α_{k}} ⊂ V. Hence, {P_{α_{i}} } is a network at x_{a} in X.

(3.2.2) K ⊂ f(L). Assume that x ∈ K. For each i ∈ ℕ, pick α_{i} ∈ Γ_{i} such that x ∈ F_{α_{i}} . Put a = (α_{i}), it follows that a ∈ L. By the proof of (3.2.1), f(a) = x. So, K ⊂ f(L).

(3.2.3) L is compact. Because each Γ_{i} is finite, ∏ _{i∈ℕ} Γ_{i} is compact. Note that L ⊂ ∏ _{i∈ℕ} Γ_{i}, we only need to prove that L is closed in ∏ _{i∈ℕ} Γ_{i}. In fact, let a = (α_{i}) ∈ ∏ _{i∈ℕ} Γ_{i} − L. Then, ⋂ _{i∈ℕ} F_{α_{i}} = ∅. From the compactness of K, there exists i_{0} ∈ ℕ such that ⋂ _{i⩽i_{0}} F_{α_{i}} = ∅. Put W = {(β_{i}) ∈ ∏ _{i∈ℕ} Γ_{i} : β_{i} = α_{i} for each i ⩽ i_{0}}. Then, W is an open subset of ∏ _{i∈ℕ} Γ_{i} satisfying a ∈ W and W ∩ L = ∅. This implies that L is a closed subset of ∏ _{i∈ℕ} Γ_{i}. Therefore, L is a compact subset of M.

(4) Let P be an sn-network. Then, X is sn-first countable. Since every snnetwork is cs-network, it follows from (3) that f is a sequence-covering, compactcovering map. By Remark 2.1(2) and [1, Proposition 2.2(1)], f is 1-sequence-covering.

(5) Let P be an so-network. Since each so-network is a cs-network, by (3), it suffices to prove that f is 2-sequence-covering.

Let x ∈ X and a = (α_{i}) ∈ f^{−1} (x). It is obvious that each P_{α_{i}} is a sequential neighborhood of x in X. For each n ∈ ℕ, put B_{n} = {(γ_{i}) ∈ M : γ_{i} = α_{i} for each i ⩽ n}. Then, {B_{n}} is a decreasing neighborhood base of a in M, and f(B_{n}) = ⋂ _{i⩽n} P_{α_{i}} for all n ∈ ℕ. Now, let {x_{n}} be a sequence converging to x in X. Since each f(B_{n}) is a sequential neighborhood at x in X, it follows from [10, Lemma 3.2] that for each n ∈ ℕ, there exists a_{n} ∈ f^{−1} (x_{n}) such that the sequence {a_{n}} converging to a in M. Therefore, f is 2-sequence-covering. □

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

(1) X has a Lindelöf cs^{∗}-network with property σ-(P);

(2) X has a Lindelöf cfp-network with property σ-(P);

(3) X has a Lindelöf cs-network with property σ-(P);

(4) X is a sequence-covering, compact-covering α(P)-image of a locally separable metric space;

(5) X is a sequentially-quotient α(P)-image of a locally separable metric space;

(6) X is a sequentially-quotient α(P)-image of a metric space, and has an so-cover consisting of H-ℵ_{0}-subspaces.

Proof. (1) ⇔ (2) ⇔ (3). By Lemma 2.1 and Lemma 2.2.

(3) ⇒ (4). By Lemma 2.4.

(4) ⇒ (5). It is obvious.

(5) ⇒ (6). Assume that (5) holds. It suffices to prove that X has an so-cover consisting of H-ℵ_{0}-subspaces. In fact, by Lemma 2.3(1) and Lemma 2.2, X has a Lindelöf cs-network P having property σ-(P). Then, each element of P is an H-ℵ_{0}-subspace. By the proof of (2) ⇒ (3) in [20, Theorem 3.4], X has an so-cover consisting of H-ℵ_{0}-subspaces.

(6) ⇒ (1). Let O be an so-cover consisting of H-ℵ_{0}-subspaces of X and f : M → X be a sequentially-quotient α(P)-map, where M is a metric space. Similar to the proof of Lemma 2.3, there exists a base B of M such that P = f(B) having property σ-(P). Since f is sequentially-quotient, P is a cs^{∗}-network for X. We can assume that P is closed under finite intersections. Let G = {P ∈ P : P ⊂ O, O ∈ O}. Then, each element of G is an H-ℵ_{0}-subspace. Hence, each element of G is Lindelöf. Now, we proved that G is a cs^{∗}-network. In fact, let L be a sequence converging to x ∈ U with U open in X. Since O is an so-cover for X, there exists O ∈ O such that x ∈ O. On the other hand, since P is a point-countable cs^{∗}-network, it follows from [25, Lemma 3] that there exists a finite subfamily H ⊂ (P)_{x} such that L is eventually in ⋃ H ⊂ U. So, the family {H ⊂ (P)_{x} : H is finite and L is eventually in ⋃ H ⊂ U} is non-empty. Furthermore, since (P)_{x} is countable, we can write {H ⊂ (P)_{x} : H is finite and L is eventually in ⋃ H ⊂ U} = {H_{n} : n ∈ ℕ}. For each n ∈ ℕ, let H_{n} = ⋂ _{i⩽n} (⋃ H_{i}). It is obvious that L is eventually in each H_{n}. Now, we shall show that H_{n} ⊂ O for some n ∈ ℕ. If not, for each n ∈ ℕ, there exists x_{n} ∈ H_{n} − O. Then, {x_{i}} converges to x. Indeed, let x ∈ W with W is open in X. Then, U ∩ W is an open neighborhood of x. By [25, Lemma 3], there exists a finite subfamily Q ⊂ (P)_{x} such that L is eventually in ⋃ Q and ⋃ Q ⊂ U ∩ W. Since Q is a finite subfamily of (P)_{x} and L is eventually in ⋃ Q ⊂ U, Q = H_{n} for some n ∈ ℕ. Furthermore, since x_{i} ∈ H_{i} for all i ∈ ℕ and

H_{i} = ⋂_{j⩽i} ( ⋃ H_{j} ) ⊂ ⋂_{j⩽n} ( ⋃ H_{j} ) ⊂ ⋃ H_{n} ⊂ W,

for all i ⩾ n, we get x_{i} ∈ W for all i ⩾ n. Therefore, {x_{i}} converges to x. Since O is a sequential neighborhood of x, this implies that there exists n ∈ ℕ such that x_{i} ∈ O for all i ⩾ n. This is a contradiction to x_{i} ∉ O for all i ∈ ℕ. Thus, H_{n} ⊂ O for some n ∈ ℕ.

On the other hand, since H_{n} = ⋂ _{i⩽n} ( ⋃ H_{i}) = ⋃ { ⋂ _{i⩽n} F_{i} : F_{i} ∈ H_{i}}, and L is eventually in H_{n}, it implies that for each i ⩽ n, there exists F_{i} ∈ H_{i} such that L is frequently in F = ⋂ _{i⩽n} F_{i}. Since P is closed under finite intersections, F ∈ P. Then, L is frequently in F, F ⊂ U and F ∈ G. Thus, G is a cs^{∗}-network for X. By Remark 2.1(1), G is a Lindelöf cs^{∗}-network having property σ-(P). □

Remark 2.2. By Theorem 2.1, in case that the property (P) is locally countable, we get an affirmative answer to Question 2.17 of [3].

By Theorem 2.1, the following corollary holds.

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

(1) X is a k-space with a Lindelöf cs^{∗}-network having property σ-(P);

(2) X is a k-space with a Lindelöf cfp-network having property σ-(P);

(3) X is a k-space with a Lindelöf cs-network having property σ-(P);

(4) X is a sequence-covering, compact-covering, quotient α(P)-image of a locally separable metric space;

(5) X is a quotient α(P)-image of a locally separable metric space;

(6) X is a local H-ℵ_{0}-space and a quotient α(P)-image of a metric space.

Remark 2.3. By Corollary 2.1, we get an affirmative answer to the Question 1.2.

Remark 2.4. Let P be a network having property σ-(P) for a regular space X. Then,

(1) If P is a cs^{∗}-network (cfp-network; cs-network), then P is Lindelöf iff each element of P is a cosmic subspace, iff each element of P is a ℵ_{0}-subspace.

(2) If P is an sn-network, then P is Lindelöf iff each element of P is a cosmic subspace, iff each element of P is an sn-second countable subspace.

(3) If P is an so-network, then P is Lindelöf iff each element of P is a cosmic subspace, iff each element of P is an so-second countable subspace.

By Theorem 2.1 and Remark 2.4, we obtain the following results for Nguyen Van Dung in case X is a regular space.

Corollary 2.2. [3, Theorem 2.8], The following are equivalent for a regular space X.

(1) X has a σ-locally countable cs-network consisting of ℵ_{0}-subspaces;

(2) X has a σ-locally countable cs-network consisting of cosmic subspaces;

(3) X is a sequence-covering msss-image of a locally separable metric space.

Corollary 2.3. [4, Theorem 2.1], The following are equivalent for a regular space X.

(1) X has a σ-locally finite cs-network consisting of ℵ_{0}-subspaces;

(2) X has a σ-locally finite cs-network consisting of cosmic subspaces;

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

The following results hold by means of the above results.

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

(1) X has a Lindelöf sn-network with property σ-(P);

(2) X is a 1-sequence-covering, compact-covering α(P)-image of a locally separable metric space;

(3) X is a 1-sequence-covering α(P)-image of a locally separable metric space;

(4) X is a 1-sequence-covering α(P)-image of a metric, and has an so-cover consisting of H-ℵ_{0}-subspaces.

Corollary 2.4. The following are equivalent for a space X.

(1) X has a Lindelöf weak base with property σ-(P);

(2) X is a weak-open, compact-covering α(P)-image of a locally separable metric space;

(3) X is a weak-open α(P)-image of a locally separable metric space;

(4) X is a local H-ℵ_{0}-space and a weak-open α(P)-image of a metric.

By Theorem 2.2 and Remark 2.4, we obtain the following results for Nguyen Van Dung in case X is a regular space.

Corollary 2.5. [3, Theorem 2.11] The following are equivalent for a regular space X.

(1) X has a σ-locally countable sn-network consisting of sn-second countable subspaces;

(2) X has a σ-locally countable sn-network consisting of cosmic subspaces;

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

Corollary 2.6. [4, Theorem 2.2] The following are equivalent for a regular space X.

(1) X has a σ-locally finite sn-network consisting of sn-second countable subspaces;

(2) X has a σ-locally finite sn-network consisting of cosmic subspaces;

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

Remark 2.5. By Theorem 2.2, it is possible to add the prefix “compactcovering” before “1-sequence-covering” in Corollary 2.5(3) and Corollary 2.6(3).

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

(1) X has a Lindelöf so-network with property σ-(P);

(2) X is a 2-sequence-covering, compact-covering α(P)-image of a locally separable metric space;

(3) X is a 2-sequence-covering α(P)-image of a locally separable metric space;

(4) X is a 2-sequence-covering α(P)-image of a metric, and has an so-cover consisting of H-ℵ_{0}-subspaces.

Corollary 2.7. The following are equivalent for a space X.

(1) X has a Lindelöf base with property σ-(P);

(2) X is an open, compact-covering α(P)-image of a locally separable metric space; (3) X is an open α(P)-image of a locally separable metric space;

(4) X is a local H-ℵ_{0}-space and an open α(P)-image of a metric.

By Theorem 2.3 and Remark 2.4, we obtain the following results for Nguyen Van Dung in case X is a regular space.

Corollary 2.8. [3, Theorem 2.14] The following are equivalent for a regular space X.

(1) X has a σ-locally countable so-network consisting of so-second countable subspaces;

(2) X has a σ-locally countable so-network consisting of cosmic subspaces;

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

Corollary 2.9. [4, Theorem 2.3], The following are equivalent for a regular space X.

(1) X has a σ-locally finite so-network consisting of so-second countable subspaces;

(2) X has a σ-locally finite so-network consisting of cosmic subspaces;

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

Remark 2.6. By Theorem 2.3, it is possible to add the prefix “compactcovering” before “2-sequence-covering” in Corollary 2.8(3) and Corollary 2.9(3).

3. Examples

Example 3.1. A quotient s-image of a locally separable metric space need not be locally separable (see [11, Example 9.8] or [16, Example 2.9.27]). Then, Question 1.1 is not true in the case Φ-property is an ℵ_{0}-space (or locally separable).

Example 3.2. There exists a space X with a σ-locally finite compact k-network (hence, X has a σ-locally finite Lindelöf cs-network by Theorem 2.1), but X is not locally Lindelöf (hence, X has no locally countable network) (see [24, Example 4.1(2)]). Then,

(1) A space X has a Lindelöf cs-network with property σ-(P) need not have a locally countable cs-network.

(2) In Theorem 2.1(6), X need not be local ℵ_{0}-space.

Example 3.3. S_{ω} is a Fréchet and ℵ_{0}-space, but it is not first countable. Then, it has a σ-locally finite Lindelöf cs-network. Since S_{ω} is not first countable, it doesn’t have a σ-locally countable sn-network (or weak base).

(1) A space with a σ-locally finite (hence, σ-locally countable) Lindelöf csnetwork need not have a σ-locally finite (or σ-locally countable) Lindelöf sn-network.

(2) A k-space with a σ-locally finite (hence, σ-locally countable) Lindelöf csnetwork need not have a σ-locally finite (or σ-locally countable) Lindelöf weak base.

Example 3.4. There exists a g-second countable space X, but it is not Fréchet (see, [23, Example 2.1]). Then, X has a σ-locally finite Lindelöf weak base. Since X is sequential and it is not Fréchet, X does not have a σ-locally countable so-network (or weak base). Therefore,

(1) A space with a σ-locally finite (hence, σ-locally countable) Lindelöf snnetwork need not have a σ-locally finite (or σ-locally countable) so-network.

(2) A space with a σ-locally finite (hence, σ-locally countable) Lindelöf weak base need not have a σ-locally finite (or σ-locally countable) base.

Example 3.5. There exists a space X having a locally countable sn-network, which is not an ℵ-space (see [5, Example 2.19]). Then, X has a σ-locally countable Lindelöf sn-network. Therefore,

(1) A space with a locally countable sn-network need not have a σ-locally finite Lindelöf cs-network.

(2) A space with a σ-locally countable Lindelöf sn-network need not have a σ-locally finite Lindelöf sn-network (or cs-network).

(3) A space with a σ-locally countable Lindelöf cs-network need not have a σ-locally finite Lindelöf cs-network.

Example 3.6. Using [7, Example 3.1], it is easy to see that X is Hausdorff, non-regular and X has a countable base, but it is not a sequentially-quotient πimage of a metric space. Then, X is not an ℵ_{0}-space. By Theorem 2.3, X is a 2-sequence-covering (and open) mssc-image of a locally separable metric space.

(1) There exists an H-ℵ_{0}-space, but it is not an ℵ_{0}-space.

(2) A space with a σ-locally finite Lindelöf cs-network (or an sn-network, or an so-network) need not be a sequentially-quotient π, mssc-image (or msss-image) of a metric space.

Acknowledgements. The authors would like to thank the referee for his/her valuable comments and suggestions.

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Department of Mathematics (Received 12 04 2011) Vinh University (Revised 17 07 2012) Vinh City Vietnam

    andhv@yahoo.com

Department of Mathematics

Da Nang University

Danang City Vietnam

    luongtuyench12@yahoo.com