PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 179–191 DOI: 10.2298/PIM150219022D

FURTHER RESULTS ON IMAGES OF METRIC SPACES

Nguyen Van Dung

Abstract. We introduce the notion of an ls-π-Ponomarev-system to give necessary and sufficient conditions for f : (M, M_{0}) → X to be a strong wcmapping (wc-mapping, wk-mapping) where M is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong wc-images (wc-images, wk-images) of locally separable metric spaces by means of certain networks. Also, we give counterexamples to sharpen some results on images of locally separable metric spaces in the literature.

1. Introduction

Characterizing images of metric spaces by spaces with certain networks is an interesting work in general topology [17,25]. In this field, the notion of a k-network plays an important place [27,28]. There are many modifications of k-networks such as cs-networks, cs^{∗}-networks, wcs^{∗}-networks, sn-networks, so-networks, etc. These notions are used to characterize continuous images of metric spaces with certain covering-properties [2,5,7,10,19,21,26–29]. In 2002, Lin [19] posed the following problem.

Problem. [19] Find a nice mapping such that a space with a point-countable wcs^{∗}-network can be characterized by the image of a metric space under this mapping.

This problem has been studied by some authors in Theory of k-networks. The difficulty in solving this problem is that each member P of a wcs^{∗}-network P may not contain the limit point of certain sequence which is frequently in P. Around this problem, Cai and Li [5] established a relation between spaces with point-countable wcs^{∗}-networks of certain properties and images of locally separable metric spaces; Lin and Li [20] introduced the concepts of wks-mappings and wcs-mappings, and them to characterize spaces with point-countable k-networks. Their main result is as follows.

2010 Mathematics Subject Classification: Primary 54E35, 54E40; Secondary 54D99, 54E99.

Key words and phrases: metric space, point-countable, wcs^{∗}-network.

Communicated by Miloš Kurilić.

Theorem 1.1. [20, Lemma 14] (1) A space with a point-countable k-network is a weakly continuous wks-image of a metric space.

(2) A space with point-countable wcs^{∗}-network is a weakly continuous wcs-image of a metric space.

By this result, Lin and Li characterized spaces with point-countable wcs^{∗}- network by weakly continuous wcs-images of metric spaces in [20, Theorem 15], which give a solution to Problem 1.

By the above, it is natural to ask the following questions.

Questions. (1) What are characterizations of wks-images and wcs-images of locally separable metric spaces by spaces with certain networks?

(2) Can “weakly continuous” in Theorem 1.1 be replaced by “continuous”?

We introduce the ls-π-Ponomarev-system (f, M, M_{0}, X, P) to give necessary and sufficient conditions for f : (M, M_{0}) → X to be a strong wc-mapping (wcmapping, wk-mapping) where M is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong wc-images (wcimages, wk-images) of locally separable metric spaces by means of certain networks, which implies an answer to Question (1). Also, we give counterexamples to sharpen some results on images of locally separable metric spaces in [5, 20], which give a negative answer for Question (2) and some more.

All spaces are Hausdorff, all mappings are onto, ℕ denotes the set of all natural numbers, ω = ℕ ∪ {0}, and a convergent sequence includes its limit point. Let f : X → Y be a mapping, P a family of subsets of X, and K a subset of X; we write f(P) = {f(P) : P ∈ P}, ⋃ P = ⋃ {P : P ∈ P}, ⋂ P = ⋂ {P : P ∈ P}, P|_{K} = {P ∩ K : P ∈ P}. For terms not defined here, please refer to [11] and [28].

2. The π-Ponomarev-system and images of locally separable metric spaces

First, we recall some basic notions. Let S = {x_{n} : n ∈ ℕ} be a sequence converging to x in X, and P be a subset of X. We say that S (or S ∪ {x}) is eventually in P [28], if there exists n_{0} ∈ ℕ such that {x_{n} : n ≥ n_{0}} ∪ {x} ⊂ P. Also, S (or S∪{x}) is frequently in P [28] if there exists a subsequence {x_{k_{n}} : n ∈ ℕ} of {x_{n} : n ∈ ℕ} such that {x_{k_{n}} : n ∈ ℕ} ∪ {x} ⊂ P.

Now, we present the notion of a network and their modifications. Let P be a family of subsets of a space X, a point x ∈ X, and K be a subset of X. P is a network at x in X [23], if x ∈ P for every P ∈ P, and whenever x ∈ U with U open in X, then there exists P ∈ P such that x ∈ P ⊂ U. P is a network for X [23], if for each x ∈ X, there exists P(x) ⊂ P such that P(x) is a network at x in X. P is a strong network for X [9] if for each x ∈ X, there exists P(x) ⊂ P such that P(x) is a countable network at x in X. P is a k-network for X [24], if for each compact subset K of X and K ⊂ U with U open in X, there exists a finite F ⊂ P such that K ⊂ ⋃ F ⊂ U. P is a cs-network for X [16] (resp., cs^{∗}-network for X [12]), if for each sequence S converging to x ∈ U with U open in X, there exists P ∈ P such that S ∪ {x} is eventually (resp., frequently) in P ⊂ U.

P is a wcs-network (resp., wcs^{∗}-network) for K in X, if for each sequence S in K converging to x ∈ U with U open in X, there exists P ∈ P such that S is eventually (resp., frequently) in P ⊂ U. If K = X, then P is a wcs-network for X (resp., wcs^{∗}-network for X [22]). P is a strong wcs-network (resp., strong wcs^{∗}-network) for X, if for each convergent sequence S in X, there exists P_{S} ⊂ P such that P_{S} is a countable wcs-network (resp., wcs^{∗}-network) for S in X. P is a π-network at x in X [20], if for each neighborhood U of x, there exists P ∈ P such that P ⊂ U. P is a strong π-network for X, if for each x ∈ X, there exists P(x) ⊂ P such that P(x) is a countable π-network at x in X.

Remark 2.1. (1) If P is a strong wcs^{∗}-network for X, then for each x ∈ X, there exists P(x) ⊂ P such that P(x) is a countable π-network at x in X and P(x) has the finite intersection property.

(2) If P is a network at x in X, then P is a π-network at x in X and x = ⋂ P. It implies that P has the finite intersection property.

Let f : X → Y be a mapping, and X_{0} be a subspace of X. f is an s-mapping [3] (resp., a compact mapping [4]), if f is continuous and for each y ∈ Y, f^{−1} (y) is a separable (resp., compact) subset of X. f is continuous about X_{0} [30], if for each x ∈ X and each neighborhood U of f(x) in Y there exists a neighborhood V of x in X such that f(V ∩ X_{0}) ⊂ U. f : (X, X_{0}) → Y is a ws-mapping [20], if the restriction f_{0} = f|_{X_{0}} : X_{0} → Y is an s-mapping and f is continuous about X_{0}. f is a semi-quotient mapping [20], if for each subset T of Y, T is closed when and only when \overline{f_{0}^{−1} (T)} ⊂ f^{−1} (T). f is a weakly continuous mapping [18], if f^{−1} (U) ⊂ Int[f^{−1} (\overline{U})] for each open subset U of Y.

Remark 2.2. (1) Every continuous mapping is a weakly continuous mapping.

(2) There exists a weakly continuous mapping which is not continuous, see [18, Example 3].

Lemma 2.1. [20, Lemma 1] Suppose that f : X → Y and X_{0} ⊂ X. Then the following statements are equivalent.

(1) f is continuous about X_{0}.

(2) If a net {x_{d}}_{d∈D} in X_{0} converges to a point x in X, then a net {f(x_{d})}_{d∈D} converges to f(x) in Y.

(3) If T is a subset of Y, then \overline{f_{0}^{−1} (T)} ⊂ f^{−1} (\overline{T}), where f_{0} = f|_{X_{0}} .

Remark 2.3. It follows from Lemma 2.1 that a weakly continuous mapping f : (X, X_{0}) → Y preserves the convergence of a sequence {x_{n} : n ∈ ℕ} ⊂ X_{0} which converges to x ∈ X.

Some properties of weakly continuous mappings are shown in the following lemma.

Lemma 2.2. [20] Let f : X → Y be a mapping.

(1) f is weakly continuous if and only if for each x ∈ X and each neighborhood U of f(x) in Y, there exists a neighborhood V of x in X with f(V) ⊂ \overline{U}.

(2) If f is continuous about X_{0}, then the restriction f|\overline{X_{0}} is weakly continuous.

Now we recall some notions of mappings in [20]. Let f : (X, X_{0}) → Y be a mapping which is continuous about X_{0}. f : (X, X_{0}) → Y is a wk-mapping [20] if for each compact subset K of Y and for each sequence T in K, there exists a sequence S in X_{0} such that S has an accumulation in X and f(S) is a subsequence of T. f : (X, X_{0}) → Y is a wc-mapping [20] if for each convergent sequence T in Y, there exists a sequence S in X_{0} such that S has an accumulation x in X and f(S ∪ {x}) is a subsequence of T. f : (X, X_{0}) → Y is a wcs-mapping (resp., wks-mapping) [20] if it is a ws-mapping and a wc-mapping (resp., wk-mapping).

In the same way, we call f : (X, X_{0}) → Y is a strong wc-mapping if for each convergent sequence T in Y, there exists a sequence S in X_{0} such that S has an accumulation x in X and f(S ∪{x}) = T. f : (X, X_{0}) → Y is a strong wcs-mapping if it is a ws-mapping and a strong wc-mapping.

Let P be a family of subsets of space X. P is point-countable (resp., pointfinite) [15], if for each x ∈ X, {P ∈ P : x ∈ P } is countable (resp., finite).

It is similar in spirit to [20, Lemma 13] that the following lemma holds.

Lemma 2.3. Let B be a point-countable base for a space X.

(1) If f : (X, X_{0}) → Y is a wks-mapping, then f(B|_{X_{0}} ) is a point-countable k-network for Y.

(2) If f : (X, X_{0}) → Y is a wcs-mapping, then f(B|_{X_{0}} ) is a point-countable wcs^{∗}-network for Y.

(3) If f : (X, X_{0}) → Y is a strong wcs-mapping, then f(B|_{X_{0}} ) is a pointcountable wcs-network for Y.

Let P be a strong network for a space X. We may assume that P is closed under finite intersections. Put P = {P_{α} : α ∈ A}. For every n ∈ ℕ, put A_{n} = A and endow A_{n} with a discrete topology. Put

M = {a = (α_{n}) ∈ _{∏ n∈ℕ} A_{n} : {P_{α_{n}} : n ∈ ℕ} forms a network at some x_{a} in X}.

Then M, which is a subspace of the product space ∏ _{n∈ℕ} A_{n}, is a metric space, x_{a} is unique, and x_{a} = ⋂ _{n∈ℕ} P_{α_{n}} for every a ∈ M. We define f : M → X by setting f(a) = x_{a} for every a ∈ M. The system (f, M, X, P) is called a Ponomarev-system [13, Definition 2.2]. This notion plays an important role in characterizing images of metric spaces by spaces with certain networks. In the spirit of Ponomarev-system (f, M, X, P) and the proof of [20, Lemma 14], we introduce the π-Ponomarev-system (f, M, M_{0}, X, P) as follows.

Let P be a strong wcs^{∗}-network for a space X. Assume that P is closed under finite intersections. Put P = {P_{α} : α ∈ A}. For every n ∈ ℕ, put A_{n} = A and endow A_{n} with a discrete metric. Put

M = {a = (α_{n}) ∈ _{∏ n∈ℕ} A_{n} : {P_{α_{n}} : n ∈ ℕ} has the finite intersection property

and it forms a π-network at some x_{a} in X}.

Then M, which is a subspace of the product space ∏ _{n∈ℕ} A_{n}, is a metric space, x_{a} is unique. We define f : M → X by setting f(a) = x_{a} for every a ∈ M. Put

M_{0} = {a = (α_{n}) ∈ _{∏ n∈ℕ} A_{n} : {P_{α_{n}} : n ∈ ℕ} forms a network at some x_{a} in X}.

The system (f, M, M_{0}, X, P) is called a π-Ponomarev-system.

Remark 2.4. For a π-Ponomarev-system (f, M, M_{0}, X, P), notations in the above definition are used in what follows unless otherwise specified. Moreover, we have

(1) M_{0} ⊂ M.

(2) (f, M_{0}, X, P) is a Ponomarev-system, then x_{a} = ⋂ _{n∈ℕ} P_{α_{n}} and f : M_{0} → X is continuous and onto.

For a π-Ponomarev-system (f, M, M_{0}, X, P), we have the following.

Lemma 2.4. Let (f, M, M_{0}, X, P) be a π-Ponomarev-system. Then the following statements hold.

(1) f : M → X is weakly continuous.

(2) f : (M, M_{0}) → X is continuous about M_{0}.

Proof. (1) Let a = (α_{n}) ∈ M and U be an open neighborhood of f(a) in X. Then there exists k ∈ ℕ such that P_{α_{k}} ⊂ U. Put V = {b = (β_{n}) ∈ M : β_{k} = α_{k}}. Then V is an open neighborhood of a in M. We shall prove that f(V ∩M_{0}) ⊂ U. In fact, if b = (β_{n}) ∈ V ∩ M_{0}, then β_{k} = α_{k} and f(b) ∈ ⋂ _{n∈ℕ} P_{β_{n}} ⊂ P_{β_{k}} = P_{α_{k}} ⊂ U.

(2) Using the notations in (1) again, we shall prove that f(V) ⊂ \overline{U}; then f is weakly continuous by Lemma 2.2. In fact, for each c = (γ_{n}) ∈ V and each open neighborhood W of f(c) in X, we have that P_{γ_{i}} ⊂ W for some i ∈ ℕ. So

W ∩ U ⊃ P_{γ_{i}} ∩ P_{α_{k}} = P_{γ_{i}} ∩ P_{γ_{k}} ≠ ∅.

It implies that f(c) ∈ \overline{U}. □

The next technical lemma plays an important role in the arguments.

Lemma 2.5. Let (f, M, M_{0}, X, P) be a π-Ponomarev-system. For each a = (α_{n}) ∈ M_{0} and n ∈ ℕ, put

B_{a,n} = {b = (β_{i}) ∈ M_{0} : β_{i} = α_{i} if i ⩽ n},
B_{a} = {B_{a,n} : n ∈ ℕ}, B_{n} = {B_{a,n} : a ∈ M_{0}}, B = ⋃ {B_{n} : n ∈ ℕ}.

Then the following statements hold.

(1) B_{a} is a base at a in M_{0}, and B_{n} is discrete.

(2) f(B_{a,n} ) = ⋂ _{i=1}^{n} P_{α_{i}} , and f(B) = P.

Proof. (1). This is clear.

(2). For each n ∈ ℕ, let x ∈ f(B_{a,n} ). Then x = f(b) for some b = (β_{i}) ∈ B_{a,n} . This implies that x = ⋂ _{i∈ℕ} P_{β_{i}} ⊂ ⋂ _{i=1}^{n} P_{β_{i}} = ⋂ _{i=1}^{n} P_{α_{i}} . Then f(B_{a,n} ) ⊂ ⋂ _{i=1}^{n} P_{α_{i}} .

Conversely, let x ∈ ⋂ _{i=1}^{n} P_{α_{i}} , where x = f(b) for some b = (β_{i}) ∈ M. For each i ∈ ℕ, there exists γ_{n+i} ∈ A_{n+i} such that γ_{n+i} = β_{i}. Put c = (γ_{i}), where γ_{i} = α_{i} if i ⩽ n. Then we get c ∈ B_{a,n} and f(c) = x. This implies that ⋂ _{i=1}^{n} P_{α_{i}} ⊂ f(B_{a,n} ).

By above inclusions, we get f(B_{a,n} ) = ⋂ _{i=1}^{n} P_{α_{i}} . It follows from P being closed under finite intersections that f(B) = P. □

By Remark 2.4.(2) and [13, Proposition 2.8], we have the following proposition.

Proposition 2.1. Let (f, M, M_{0}, X, P) be a π-Ponomarev-system. Then f : M_{0} → X is an s-mapping (resp., a compact mapping) if and only if P is pointcountable (resp., point-finite).

In [13,14], the necessary and sufficient conditions for f to be an s-mapping with covering-properties have been obtained in a Ponomarev-system (f, M, X, P) where P is some network except for a point-countable wcs^{∗}-network (k-network, wcsnetwork). The reason of this exception is that each member P of a wcs^{∗}-network (k-network, wcs-network) P may not contain the limit point of certain sequence which is frequently in P, then the mapping f may not be a covering-mapping. In hat follows, by means of wcs-network (wcs^{∗}-network, k-network), we obtain the necessary and sufficient conditions for f : (M, M_{0}) → X to be a strong wc-mapping (wc-mapping, wk-mapping) in a π-Ponomarev-system (f, M, M_{0}, X, P).

Lemma 2.6. [8, Lemma 2.6] Let P be a cs-network for X and S be a convergent sequence such that S ⊂ U with U open in X. Then there exists F ⊂ P satisfying

(1) F is finite.

(2) For each F ∈ F, ∅ ≠ F ∩ S ⊂ F ⊂ U.

(3) For each x ∈ S, there exists a unique F ∈ F such that x ∈ F.

(4) If F ∈ F contains the limit point of S, then S − F is finite.

Such an F is called to have property cs(S, U).

Theorem 2.1. Let (f, M, M_{0}, X, P) be a π-Ponomarev-system. Then the following statements hold.

(1) f : (M, M_{0}) → X is a strong wc-mapping if and only if P is a strong wcs-network for X.

(2) f : (M, M_{0}) → X is a wc-mapping if and only if P is a strong wcs^{∗}- network for X.

(3) f : (M, M_{0}) → X is a wk-mapping if and only if P is a strong k-network for X.

Proof. (1) Necessity. Let f be a strong wc-mapping. For each convergent sequence S of X, there exists a sequence L of M_{0} which has an accumulation a_{0} in M and f(L ∪ {a_{0}}) = S. By using notations in Lemma 2.5 again, we have that B is a base for M_{0}. In spirit of the proof of Lemma 2.3(3), P = f(B) is a strong wcs-network for X.

Sufficiency. Let P be a strong wcs-network for X. We shall prove that f : (M, M_{0}) → X is a strong wc-mapping by the following two claims.

Claim 1. f : (M, M_{0}) → X is continuous about M_{0}.

It follows from Lemma 2.4.

Claim 2. f : (M, M_{0}) → X is a strong wc-mapping.

For each convergent sequence S = {x_{m} : m ∈ ω} converging to x_{0} in X. We may assume that x_{n} ≠ x_{m} for every n ≠ m. Since P is a strong wcs-network for X, there exists P_{S} ⊂ P such that P_{S} is a countable wcs-network for S in X. Put P_{S}^{0} = {P ∈ P_{S} : S is eventually in P }. Then P_{S}^{0} is countable. Pick some P_{0} ∈ P_{S}^{0}, we have that S − P_{0} is finite. For each x ∈ S − P_{0}, there exists P_{x} ∈ P_{S} such that P_{x} ⊂ X − (S − {x}). Put

F = {P_{x} : x ∈ S − P_{0}} ∪ {P_{0}}, F^{0} = {P_{x} : x ∈ S − P_{0}} ∪ {P_{0} ∪ {x_{0}}}.

Then F^{0} has the property cs(S, X) (see Lemma 2.6). It implies that

{F ⊂ P_{S} : F^{0} has property cs(S, X)} ≠ ∅.

So we can put {F ⊂ P_{S} : F^{0} has property cs(S, X)} = {F_{n} : n ∈ ℕ}.

For each n ∈ ℕ, put F_{n} = {P_{α} : α ∈ B_{n}}, where B_{n} is a finite subset of A_{n}. For each n ∈ ℕ and m ∈ ℕ, by the definition of property cs(S, X), there exists unique α_{m,n} ∈ B_{n} such that x_{m} ∈ P_{α_{m,n}} ∈ F_{n}, and unique α_{0,n} ∈ A_{n} such that S is eventually in P_{α_{0,n}} . Put a_{m} = (α_{m,n} ) for every m ∈ ω, then a_{m} ∈ M_{0} for every m ∈ ℕ and a_{0} ∈ M. We shall prove that a_{m} → a_{0} in M. In fact, for each n ∈ ℕ, by the definition of property cs(S, X), there exists k_{n} ∈ ℕ such that x_{m} ∈ P_{α_{0,n}} for every m ≥ k_{n}. Then α_{m,n} = α_{0,n} for every m ≥ k_{n}. It implies that a_{m} → a_{0} in M. To complete the proof, we need only to prove f(a_{m}) = x_{m} for every m ∈ ℕ. Let x_{m} ∈ U with U open in X. Then S − {x_{m}} is a convergent sequence converging to x_{0} in X and S − {x_{m}} ⊂ X − {x_{m}} where X − {x_{m}} is open in X. By Lemma 2.6, there exists F ⊂ P_{S} such that F^{0} has the property cs(S − {x_{m}}, X − {x_{m}}). Since U − (S − {x_{m}}) is an open neighborhood of x_{m} in X, there exists P_{m} ∈ P_{S} such that x_{m} ∈ P_{m} ⊂ U − (S − {x_{m}}). Then F^{0} ∪ {P_{m}} has the property cs(S, X). There exists k ∈ ℕ such that F^{0} ∪ {P_{m}} = F_{k}^{0}; it means F ∪ {P_{m}} = F_{k}. Then x_{m} ∈ P_{α_{m,k}} = P_{m} ⊂ U. It implies that {P_{α_{m,n}} : n ∈ ℕ} forms a network at x_{m} in X, so f(a_{m}) = x_{m} for every m ∈ ℕ.

(2) Same as the proof of (1).

(3) Necessity. Same as the necessity of (1).

Sufficiency. By the sufficiency of (1), we only need to prove that f is a wkmapping. For a compact subset K of X and each sequence T = {x_{n} : n ∈ ℕ} in K, we can assume that the sequence {x_{n} : n ∈ ℕ} converges to a point x ∈ K − {x_{n} : n ∈ ℕ} by [20, Lemma 12]. It follows from the proof of Claim 2 in the sufficiency of (1) above, there exists a sequence S in M such that S converges to x in M and f(S ∪ {x}) is a subsequence of T. Then f is a wk-mapping. □

From Proposition 2.1 and Theorem 2.1, we have the following corollary.

Corollary 2.1. Let (f, M, M_{0}, X, P) be a π-Ponomarev-system. Then the following statements hold.

(1) f : (M, M_{0}) → X is a strong wcs-mapping if and only if P is a pointcountable wcs-network for X.

(2) f : (M, M_{0}) → X is a wcs-mapping if and only if P is a point-countable wcs^{∗}-network for X.

(3) f : (M, M_{0}) → X is a wks-mapping if and only if P is a point-countable k-network for X.

In [20, Lemma 14], Lin and Li proved that spaces with point-countable knetworks (resp., wcs^{∗}-network) are weakly continuous wks-(resp., wcs-)images of metric spaces. By using their method, we characterize the strong wcs-image of a metric space by a space with a point-countable wcs-network as follows.

Proposition 2.2. A space has a point-countable wcs-network if and only if it is a weakly continuous strong wcs-image of a metric space.

In the following, we prove that “weakly continuous” in the result of Lin and Li, and Proposition 2.2 can not be replaced by “continuous”.

Lemma 2.7. [1, Lemma 12(3)] Let f : X → Y be a sequentially-quotient mapping. If P is a cs^{∗}-network for X, then f(P) is a cs^{∗}-network for Y.

Lemma 2.8. Let f : (X, X_{0}) → Y be a weakly continuous wcs-mapping and X be a metric space. If f is continuous, then Y has a point-countable cs^{∗}-network.

Proof. Let B be a σ-locally finite base of X. Then C = B|\overline{X_{0}} is a σlocally finite base of \overline{X_{0}}. Since f : (X, X_{0}) → Y is a wcs-mapping, f|\overline{X_{0}} is a sequentially-quotient mapping. Put P = f|\overline{X_{0}} (B|\overline{X_{0}} ), then P is a cs^{∗}-network for Y by Lemma 2.7. On the other hand, B|\overline{X_{0}} = B|_{X_{0}} and f|_{X_{0}} is an s-mapping, then P is point-countable. It implies that P is a point-countable cs^{∗}-network for Y. □

Example 2.1. There exists a space which has a point-countable wcs-network and k-network, and has not any point-countable cs^{∗}-network.

Proof. Let S_{ω_{1}} be the quotient space of the sum of uncountably many convergent sequences S_{λ} = {x_{λ,n} : n ∈ ω}, λ ∈ Λ, by identifying all limit points x_{λ,0} to a point x_{0}. Then S_{ω_{1}} does not have any point-countable cs^{∗}-network. Put P = {{x_{0}}} ∪ {{x_{λ,i} : i ≥ n} : n ∈ ℕ, λ ∈ Λ}. We have that P is a point-countable wcs-network and k-network for X. □

Remark 2.5. By Example 2.1, S_{ω_{1}} does not have any point-countable cs^{∗}- network. Then S_{ω_{1}} is not any continuous wcs-image of a metric space by Lemma 2.8. Note that S_{ω_{1}} has a point-countable wcs-network, then it has a point-countable wcs^{∗}-network. Also, S_{ω_{1}} has a point-countable k-network. Therefore, “weakly continuous” in the result of Lin and Li, and Proposition 2.2 can not be replaced by “continuous”.

3. Further results on images of locally separable metric spaces

A space X is an ℵ_{0}-space if X is regular and it has a countable cs-network [23]. Cai and Li [5] established the mapping relation between a space with a pointcountable wcs^{∗}-network and a locally separable metric space. The main result is as follows.

Proposition 3.1. [5, Theorem 2.6] If a space X has a point-countable wcs^{∗}- network P and the closure of each element of P is an ℵ_{0}-subspace, then X is the pseudo-sequence-covering image of a locally separable metric space.

By this result, it is natural to ask whether the inverse implication is true? The answer is negative by the following Example 3.1. Note that every sequence-covering mapping is a pseudo-sequence-covering mapping.

Example 3.1. There exists a sequence-covering image of a locally separable metric space which does not have any point-countable wcs^{∗}-network.

Proof. Consider the butterfly space Y of McAuley [23, page 999], which is defined as follows: Let Y be the upper half-plane, and let A ⊂ Y denote the x-axis. Points in Y − A have ordinary plane neighborhoods. A base for the neighborhoods of a point p ∈ A consists of all sets N_{ε}(p) where N_{ε}(p) consists of p together with all points q ∈ Y having distance < ε from p and lying underneath the union of the two rays in Y which emanate from p and have slopes ε and −ε, respectively. Such an N_{ε}(p) is called a butterfly neighborhood of p.

It follows from [23, Example 12.1] that Y is regular and Fréchet, and not an ℵ_{0}-space. By [15, Theorem 5.2] and [28, Remark 1], Y has not any point-countable wcs^{∗}-network. On the other hand, Y is a sequence-covering image of a metric space by [6, Theorem 5]. □

In [1], an ls-Ponomarev-system (f, M, X, {P_{λ}}) was introduced to give necessary and sufficient conditions for f to be an s-mapping with covering-properties from a locally separable metric space M onto a space X with a double cover.

Let {X_{λ} : λ ∈ Λ} be a cover for a space X such that each X_{λ} has a network P_{λ} which is closed under finite intersections. {(X_{λ}, P_{λ}) : λ ∈ Λ} is a double cover for X [1], if each P_{λ} is countable. {(X_{λ}, P_{λ}) : λ ∈ Λ} is point-countable [1] (resp., point-finite), if {X_{λ} : λ ∈ Λ} is point-countable (resp., point-finite).

Let {(X_{λ}, P_{λ}) : λ ∈ Λ} be a double cover for a space X and (f_{λ}, M_{λ}, X_{λ}, P_{λ}) be a Ponomarev-system for every λ ∈ Λ. Since P_{λ} is countable, M_{λ} is a separable metric space. Put M = ⨁ _{λ∈Λ} M_{λ} and f = ⨁ _{λ∈Λ} f_{λ}. Then M is a locally separable metric space and f is a mapping from M onto X. The system (f, M, X, {P_{λ}}) is an ls-Ponomarev-system [1].

Now, we introduce a notion of an ls-π-Ponomarev-system as follows. Let {(X_{λ}, P_{λ}) : λ ∈ Λ} be a double cover for X and (f_{λ}, M_{λ}, M_{λ,0} , X_{λ}, P_{λ}) be the π-Ponomarev-system for every λ ∈ Λ. Since P_{λ} is countable, M_{λ} is a separable metric space. Put M = ⨁ _{λ∈Λ} M_{λ}, M_{0} = ⨁ _{λ∈Λ} M_{λ,0} , f = ⨁ _{λ∈Λ} f_{λ}V. Then M is a locally separable metric space and f is a continuous mapping from M onto X. The system (f, M, M_{0}, X, {P_{λ}}) is called an ls-π-Ponomarev-system.

Remark 3.1. For an ls-π-Ponomarev-system (f, M, M_{0}, X, {P_{λ}}), notations

in the above definition are used in what follows next unless otherwise specified. Moreover, we have

(1) M_{0} ⊂ M. (2) X_{λ} = f(M_{λ,0} ) = f(M_{λ}) for every λ ∈ Λ.

(3) (f, M_{0}, X, {P_{λ}}) is an ls-Ponomarev-system, then f : M_{0} → X is continu-

ous and onto.

By Lemma 2.4, we have the following lemma.

Lemma 3.1. Let (f, M, M_{0}, X, {P_{λ}}) be an ls-π-Ponomarev-system. Then the following statements hold.

(1) f : M → X is weakly continuous.

(2) f : (M, M_{0}) → X is continuous about M_{0}.

It follows from Remark 3.1 and [1, Theorem 2.14] that the following result holds.

Proposition 3.2. Suppose that (f, M, M_{0}, X, {P_{λ}}) is an ls-π-Ponomarev-system. Then f : M_{0} → X is an s-mapping (resp., a compact mapping) if and only if {(X_{λ}, P_{λ}) : λ ∈ Λ} is point-countable (resp., point-finite).

We give a necessary and sufficient condition for the mapping f : (M, M_{0}) → X to be a strong wc-mapping in an ls-π-Ponomarev-system (f, M, M_{0}, X, {P_{λ}}).

Proposition 3.3. Suppose that (f, M, M_{0}, X, {P_{λ}}) is an ls-π-Ponomarev-system. Then following statements are equivalent.

(1) f : (M, M_{0}) → X is a strong wc-mapping.

(2) For each sequence S converging to x in X, there exists λ ∈ Λ such that (a) S ∪ {x} is eventually in X_{λ}.

(b) For each open neighborhood U of x in X_{λ}, there exists P ∈ P_{λ} such that S ∩ X_{λ} is eventually in P ⊂ U.

Proof. (1) ⇒ (2). For each sequence S converging to x in X, we need only to prove the following two claims. Assume that x ∉ S, if necessary.

Claim 1. There exists λ ∈ Λ such that S ∪ {x} is eventually in X_{λ}.

Since f : (M, M_{0}) → X is a strong wc-mapping, there exists a sequence L converging to a in M such that f(L) = S and L ⊂ M_{0}. Since L ∪ {a} is a convergent sequence in M, there exists λ ∈ Λ such that L is eventually in M_{λ}. It implies that S is eventually in X_{λ}.

Claim 2. For S and λ in Claim 1, if x ∈ U with U open in X_{λ}, then there exists P ∈ P_{λ} such that S ∩ X_{λ} is eventually in P ⊂ U.

By using notations in the proof of Claim 1 again, we have that L converges to a ∈ f_{λ}^{−1} (U) with f_{λ}^{−1} (U) open in M_{λ}. Put a = (α_{i}), and

A_{a,n} = {b = (β_{i}) ∈ M_{λ} : β_{i} = α_{i} for every i ⩽ n},
B_{a,n} = {b = (β_{i}) ∈ M_{λ,0} : β_{i} = α_{i} for every i ⩽ n},

for every n ∈ ℕ. Then {A_{a,n} : n ∈ ℕ} is a base at a in M_{λ}. So, there exists n ∈ ℕ such that L ∪ {a} is eventually in A_{a,n} ⊂ f_{λ}^{−1} (U). Then L is eventually in A_{a,n} ∩ M_{λ,0} ⊂ f_{λ}^{−1} (U) ∩ M_{λ,0} . Note that A_{a,n} ∩ M_{λ,0} = B_{a,n} . So L is eventually in B_{a,n} ⊂ f_{λ}^{−1} (U). Then S is eventually in P ⊂ U where P = f_{λ}(B_{a,n} ) ∈ P_{λ} by Lemma 2.5.

(2) ⇒ (1). We need only to prove the following two claims.

Claim 1. f : (M, M_{0}) → X is continuous about M_{0}.

This follows from Lemma 3.1.

Claim 2. f : (M, M_{0}) → X is a strong wc-mapping.

Let S be a sequence converging to x in X. Then S ∪ {x} is eventually in X_{λ} for some λ ∈ Λ. Put S_{λ} = S ∩X_{λ} and H_{λ} = S −X_{λ}. Then S_{λ} ∪{x} is a convergent sequence converging to x in X_{λ} and H_{λ} is finite. As in the proof of Claim 2 in the sufficiency of Theorem 2.1(1), there exists a sequence L_{λ} ⊂ M_{λ,0} converging to a in M_{λ} such that f_{λ}(L_{λ}) = S_{λ}. On the other hand, there exists a finite subset G of M_{0} such that H_{λ} = f(G). Then L = G ∪ L_{λ} is a sequence converging to a in M, L ⊂ M_{0}, and f(L) = S. This proves that f is a strong wc-mapping. □

Being similar in spirit to the proof of Proposition 3.3, we get the following propositions which are necessary and sufficient conditions for f : (M, M_{0}) → X to be a wc-mapping (wk-mapping) in an ls-π-Ponomarev-system (f, M, M_{0}, X, {P_{λ}}).

Proposition 3.4. Suppose that (f, M, M_{0}, X, {P_{λ}}) is an ls-π-Ponomarev-system. Then the following statements are equivalent.

(1) f : (M, M_{0}) → X is a wc-mapping.

(2) For each sequence S converging to x in X, there exists λ ∈ Λ such that (a) S ∪ {x} is frequently in X_{λ}.

(b) For each open neighborhood U of x in X_{λ}, there exists P ∈ P_{λ} such that S ∩ X_{λ} is frequently in P ⊂ U.

Proposition 3.5. Suppose that (f, M, M_{0}, X, {P_{λ}}) is an ls-π-Ponomarev-system. Then the following statements are equivalent.

(1) f : (M, M_{0}) → X is a wk-mapping.

(2) For each sequence S in a compact subset K of X, there exists λ ∈ Λ such that

(a) S ∪ {x} is frequently in X_{λ}.

(b) For each open neighborhood U of x in X_{λ}, there exists P ∈ P_{λ} such that S ∩ X_{λ} is frequently in P ⊂ U.

It follows from Proposition 3.3 that we have a characterization of strong wcimages of locally separable metric spaces as follows.

Corollary 3.1. For a space X, the following statements are equivalent.

(1) X is a weakly continuous strong wc-image (resp., wcs-image) of a locally separable metric space.

(2) X has a double cover (resp., point-countable double cover) {(X_{λ}, P_{λ}) : λ ∈ Λ } satisfying for each sequence S converging to x in X there exists λ ∈ Λ such that

(a) S ∪ {x} is eventually in X_{λ}.

(b) For each open neighborhood U of x in X_{λ}, there exists P ∈ P_{λ} such that S ∩ X_{λ} is eventually in P ⊂ U.

Proof. (1) ⇒ (2). Let f : (M, M_{0}) → X be a weakly continuous strong wc-mapping where M is a locally separable metric space. By [11, 4.4.F], M = ⨁ _{λ∈Λ} M_{λ} where each M_{λ} is a separable metric space. For each λ ∈ Λ, let B_{λ} be a countable base for M_{λ}. Put P_{λ} = f(B_{λ}) and X_{λ} = f(M_{λ}). As in the proof (1)⇒(2) of Proposition 3.3, we have that {(X_{λ}, P_{λ}) : λ ∈ Λ} satisfies the required conditions.

For the parenthetic part, {(X_{λ}, P_{λ}) : λ ∈ Λ} is point-countable as in the necessity’s proof of Proposition 3.2.

(2) ⇒ (1). Let {(X_{λ}, P_{λ}) : λ ∈ Λ} be a double cover for X which satisfies conditions in statement (2). Then the ls-π-Ponomarev (f, M, M_{0}, X, {P_{λ}}) exists.

By Lemma 3.1 and Proposition 3.3, f : (M, M_{0}) → X is a weakly continuous strong wc-mapping and M is a locally separable metric space. Then X is a weakly continuous strong wc-image of a locally separable metric space.

For the parenthetic part, f : (M, M_{0}) → X is a ws-mapping by Proposition 3.2. Then X is a weakly continuous strong wcs-image of a locally separable metric space. □

Being similar in spirit to the proof of Corollary 3.1, we get the following corollaries on characterizations of weakly continuous ws-images of locally separable metric spaces.

Corollary 3.2. For a space X, the following statements are equivalent.

(1) X is a weakly continuous wc-image (resp., wcs-image) of a locally separable metric space.

(2) X has a double cover (resp., point-countable double cover) {(X_{λ}, P_{λ}) : λ ∈ Λ } satisfying for each sequence S converging to x in X, there exists λ ∈ Λ such that

(a) S ∪ {x} is frequently in X_{λ}.

(b) For each open neighborhood U of x in X_{λ}, there exists P ∈ P_{λ} such that S ∩ X_{λ} is frequently in P ⊂ U.

Corollary 3.3. For a space X, the following statements are equivalent.

(1) X is a weakly continuous wk-image (resp., wks-image) of a locally separable metric space.

(2) X has a double cover (resp., point-countable double cover) {(X_{λ}, P_{λ}) : λ ∈ Λ} satisfying for each sequence S in a compact subset K of X, there exists λ ∈ Λ such that

(a) S ∪ {x} is frequently in X_{λ}.

(b) For each open neighborhood U of x in X_{λ}, there exists P ∈ P_{λ} such that S ∩ X_{λ} is frequently in P ⊂ U.

Acknowledgment. The author would like to thank the referees for their valuable comments. Also, the author would like to thank Prof. T. V. An (Vinh University) and Prof. S. Lin (Ningde Normal University) for their excellent advices and supports.

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Institute of Research and Development (Received 26 11 2012) Duy Tan University R.809, K7/25 Quang Trung Da Nang City and Faculty of Mathematics and Information Technology Teacher Education Dong Thap University Cao Lanh City Dong Thap Province Viet Nam

    nvdung@dthu.edu.vn