PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 93 (107) (2013), 145–152 DOI: 10.2298/PIM1307145T

SPACES WITH σ-LOCALLY FINITE LINDELÖF sn-NETWORKS

Luong Quoc Tuyen

Communicated by Miloš Kurilić

Abstract. We prove that a space X has a σ-locally finite Lindelöf sn-network if and only if X is a compact-covering compact and mssc-image of a locally separable metric space, if and only if X is a sequentially-quotient π and mssc-image of a locally separable metric space, where “compact-covering” (or “sequentially-quotient”) can not be replaced by “sequence-covering”. As an application, we give a new characterization of spaces with locally countable weak bases.

1. Introduction

In [17] Lin introduced the concept of mssc-maps to characterize spaces with certain σ-locally finite networks by mssc-images of metric spaces. After that, some characterizations for certain mssc-images of metric (or semi-metric) spaces are obtained by many authors ([11, 12, 14], for example). Recently, Dung gave some characterizations for certain mssc-images of locally separable metric spaces (see in [3]).

We prove that a space X has a σ-locally finite Lindelöf sn-network if and only if X is a compact-covering compact and mssc-image of a locally separable metric space, if and only if X is a sequentially-quotient π and mssc-image of a locally separable metric space, where “compact-covering” (or “sequentially-quotient”) can not be replaced by “sequence-covering”. As an application, we give a new characterization of spaces with locally countable weak bases.

Throughout this paper, all spaces are assumed to be T_{1} and regular, all maps are continuous and onto, ℕ denotes the set of all natural numbers. Let P and Q be two families of subsets of X and x ∈ X, we denote (P)_{x} = {P ∈ P : x ∈ P }, ⋃ P = ⋃ {P : P ∈ P}, ⋂ P = ⋂ {P : P ∈ P}, st(x, P) = ⋃ (P)_{x} and P ⋀ Q = {P ∩ Q : P ∈ P, Q ∈ Q}. 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.

2010 Mathematics Subject Classification: Primary 54E35, 54E40; Secondary 54D65, 54E99.

Key words and phrases: weak base, sn-network, locally finite, Lindelöf, compact-covering map, compact map, mssc-map.

Definition 1.1. Let X be a space, P ⊂ X and let P be a cover of X.

(1) P is a sequential neighborhood of x in X [5], if each sequence S converging to x is eventually in P.

(2) P is a sequentially open subset of X [5], if P is a sequential neighborhood of x in X for every x ∈ P.

(3) P is an so-cover for X [19], if each element of P is sequentially open in X.

(4) P is a cfp-cover for X [29], if whenever K is compact subset of X, there exist a finite family {K_{i} : i ⩽ n} of closed subsets of K and {P_{i} : i ⩽ n} ⊂ P such that K = ⋃ {K_{i} : i ⩽ n} and each K_{i} ⊂ P_{i}.

(5) P is an cs^{∗}-cover for X [28], if every convergent sequence is frequently in some P ∈ P.

Definition 1.2. Let P be a family of subsets of a space X.

(1) For each x ∈ X, P is a network at x in X [18], if x ∈ ⋂ P, and if x ∈ U with U open in X, then there exists P ∈ P such that x ∈ P ∈ U.

(2) P is a cs-network for X [28], if each sequence S converging to a point 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 [28], if for each sequence S converging to a point x ∈ U with U open in X, S is frequently in P ⊂ U for some P ∈ P.

(4) P is Lindelöf, if each element of P is a Lindelöf subset of X.

(5) P is point-countable [4], if each point x ∈ X belongs to only countably many members of P.

(6) P is locally countable [4], if for each x ∈ X, there exists a neighborhood V of x such that V meets only countably many members of P.

(7) P is locally finite [4], if for each x ∈ X, there exists a neighborhood V of x such that V meets only finite many members of P.

(8) P is star-countable [23], if each P ∈ P meets only countably many members of P.

Definition 1.3. Let P = ⋃ {P_{x} : x ∈ X} be a family of subsets of a space X satisfying that, for every 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 [1], if G ⊂ X such that for every 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 [16], if each member of P_{x} is a sequential neighborhood of x for all x ∈ X. Here, P_{x} is an sn-network at x in X.

Definition 1.4. Let X be a space.

(1) X is an sn-first countable space [8], if there is a countable sn-network at x in X for all x ∈ X.

(2) X is an sn-metrizable space [7] (resp., a g-metrizable space [25]), if X has a σ-locally ŕfinite sn-network (resp., weak base).

SPACES WITH σ-LOCALLY FINITE LINDELÖF SN-NETWORKS 147

(3) X is a cosmic space [21], if X has a countable network.

(4) X is an ℵ_{0}-space [21], if X has a countable cs-network.

(5) X is an ℵ-space [22], if X has a σ-locally finite cs-network.

(6) X is a sequential space [5], if each sequentially open subset of X is open.

(7) X is a Fréchet space [4], if for each x ∈ \overline{A}, there exists a sequence in A converging to x.

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

(1) f is sequence-covering [24], 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 map is a strong sequence-covering map in the sense of [14].

(2) f is compact-covering [21], if for each compact subset K of Y, there exists a compact subset L of X such that f(L) = K.

(3) f is pseudo-sequence-covering [13], if for each convergent sequence S of Y, there exists a compact subset K of X such that f(K) = S.

(4) f is sequentially-quotient [2], if for each convergent sequence S of Y, there exists a convergent sequence L of X such that f(L) is a subsequence of S.

(5) f is a quotient map [4], if whenever U ⊂ Y, U open in Y if and only if f^{−1} (U) open in X.

(6) f is an mssc-map [17], 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 neighborhoods of y such that each \overline{p_{i}f^{−1} (V_{i})} is compact in X_{i}.

(7) f is compact [4], if each f^{−1} (y) is compact in X.

(8) f is a π-map [13], 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.

Definition 1.6. [18] Let {P_{i}} be a cover sequence of a space X. {P_{i}} is called a point-star network, if {st(x, P_{i}) : i ∈ ℕ} is a network of x for each x ∈ X.

For some undefined or related concepts, we refer the reader to [4, 13, 18].

2. Main Results

Lemma 2.1. Let f : M → X be a sequentially-quotient mssc-map, and M be a locally separable metric space. Then, X has a σ-locally finite Lindelöf cs-network.

Proof. By using the proof of (3) ⇒ (1) in [14, Theorem 4], there exists a base B of M such that f(B) is a σ-locally finite network for X. Since M is locally separable, for each a ∈ M, there exists a separable open neighborhood U_{a}. Denote

C = {B ∈ B : B ⊂ U_{a} for some a ∈ M}.

Then, C ⊂ B and C is a separable base for M. If put P = f(C), then P ⊂ f(B), and P is a σ-locally finite Lindelöf network. Since f is sequentially-quotient and C is a base for M, P is a cs^{∗}-network. Therefore, P is a σ-locally finite Lindelöf cs^{∗}-network.

Let P = ⋃ {P_{i} : i ∈ ℕ}, we can assume that P_{n} ⊂ P_{n+1} for all n ∈ ℕ. Since each element of P_{i} is Lindelöf, each P_{i} is star-countable. It follows from [23, Lemma 2.1] that for each i ∈ ℕ, P_{i} = ⋃ {Q_{i,α} : α ∈ Λ_{i}}, where Q_{i,α} is a countable subfamily of P_{i} for all α ∈ Λ_{i} and ( ⋃ Q_{i,α} ) ∩ ( ⋃ Q_{i,β} ) = ∅ for all α ≠ β. For each i ∈ ℕ and α ∈ Λ_{i}, we put R_{i,α} = { ⋃ F : F is a finite subfamily of Q_{i,α} }. Since each R_{i,α} is countable, we can write R_{i,α} = {R_{i,α,j} : j ∈ ℕ}. Now, for each i, j ∈ ℕ, put F_{i,j} = {R_{i,α,j} : α ∈ Λ_{i}}, and denote G = ⋃ {F_{i,j} : i, j ∈ ℕ}. Then, each R_{i,α,j} is Lindelöf and each family F_{i,j} is locally finite. Now, we shall show that G is a cs-network. In fact, let {x_{n}} be a sequence converging to x ∈ U with U open in X. Since P is a point-countable cs^{∗}-network, it follows from [27, 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_{i,α} , and ⋃ A ∈ R_{i,α} . Thus, ⋃ A = R_{i,α,j} for some j ∈ ℕ. Hence, ⋃ A ∈ G, and G is a cs-network. Therefore, G is a σ-locally finite Lindelöf cs-network. □

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

(1) X is an sn-metrizable space and has an so-cover consisting of ℵ_{0}-subspaces;

(2) X has a σ-locally finite Lindelöf sn-network;

(3) X is a compact-covering compact and mssc-image of a locally separable metric space;

(4) X is a pseudo-sequence-covering compact and mssc-image of a locally separable metric space;

(5) X is a subsequence-covering compact and mssc-image of a locally separable metric space;

(6) X is a sequentially-quotient π and mssc-image of a locally separable metric space.

Proof. (1) → (2). Let P = ⋃ {P_{x} : x ∈ X} be a σ-locally finite sn-network and O be an so-cover consisting of ℵ_{0}-subspaces for X. For each x ∈ X, pick O_{x} ∈ O such that x ∈ O_{x} and put G_{x} = {P ∈ P_{x} : P ⊂ O_{x}} and G = ⋃ {G_{x} : x ∈ X}. Then, G is a σ-locally finite Lindelöf sn-network for X.

(2) → (3). Let P = ⋃ {P_{x} : x ∈ X} = {P_{n} : n ∈ ℕ} be a σ-locally finite Lindelöf sn-network for X, where each P_{n} is locally finite and each P_{x} is an snnetwork at x. Since X is a regular space, we can assume that each element of P is closed. On the other hand, since each element of P_{i} is Lindelöf, each P_{i} is star-countable. It follows from [23, Lemma 2.1] that for each i ∈ ℕ, P_{i} = ⋃ {Q_{i,α} : α ∈ Φ_{i}}, where Q_{i,α} is a countable subfamily of P_{i} for all α ∈ Φ_{i} and ( ⋃ Q_{i,α} ) ∩ ( ⋃ Q_{i,β} ) = ∅ for all α ≠ β. Since each Q_{i,α} is countable, we can write Q_{i,α} = {P_{i,α,j} : j ∈ ℕ}. Now, for each i, j ∈ ℕ, put F_{i,j} = {P_{i,α,j} : α ∈ Φ_{i}}, and A_{i,j} = {x ∈ X : P_{x} ∩ F_{i,j} = ∅} and H_{i,j} = F_{i,j} ∪ {A_{i,j} }. Then, P = ⋃ {F_{i,j} : i, j ∈ ℕ}, and

(a) Each H_{i,j} is locally finite. It is obvious.

(b) Each H_{i,j} is a cfp-cover. Let K be a non-empty compact subset of X. We shall show that there exists a finite subset of H_{i,j} which forms a cfp-cover of K.

In fact, since X has a σ-locally finite sn-network, K is metrizable. On the other hand, since P_{i} is locally finite, K meets only finitely many members of P_{i}. Thus, K meets only finitely many members of H_{i,j} . Let

Γ_{i,j} = {α ∈ Φ_{i} : P_{i,α,j} ∈ H_{i,j} , P_{i,α,j} ∩ K ≠ ∅}.

For each α ∈ Γ_{i,j} , put K_{i,α,j} = P_{i,α,j} ∩ K, F_{i,j} = \overline{K − ⋃ _{α∈Γ_{i,j}} K_{i,α,j}} . It is obvious that all K_{i,α,j} and F_{i,j} are closed subset of K, and K = F_{i,j} ∪ ( ⋃ _{α∈Γ_{i,j}} K_{i,α,j} ). Now, we only need to show F_{i,j} ⊂ A_{i,j} . Let x ∈ F_{i,j} ; then there exists a sequence {x_{n}} of K −⋃ _{α∈Γ_{i,j}} K_{i,α,j} converging to x. If P ∈ P_{x} ∩H_{i,j} , then P is a sequential neighborhood of x and P = P_{i,α,j} for some α ∈ Γ_{i,j} . Thus, x_{n} ∈ P whenever n ⩾ m for some m ∈ ℕ. Hence, x_{n} ∈ K_{i,α,j} for some α ∈ Γ_{i,j} , a contradiction. So, P_{x} ∩ H_{i,j} = ∅, and x ∈ A_{i,j} . This implies that F_{i,j} ⊂ A_{i,j} and {A_{i,j} } ∪ {P_{i,α,j} : α ∈ Γ_{i,j} } is a cfp-cover of K.

(c) {H_{i,j} : i, j ∈ ℕ} is a point-star network for X. Let x ∈ U with U open in X. Then, x ∈ P ⊂ U for some P ∈ P_{x}. Thus, there exists i ∈ ℕ such that P ∈ P_{i}. Hence, there exists a unique α ∈ Φ_{i} such that P ∈ Q_{i,α} . So, P = P_{i,α,j} ∈ H_{i,j} for some j ∈ ℕ. Since P ∈ P_{x} ∩ H_{i,j} , x ∉ A_{i,j} . Noting that P ∩ P_{i,α,j} = ∅ for all j ≠ i. Therefore, st(x, H_{i,j} ) = P ⊂ U.

Next, we write {H_{m,n} : m, n ∈ ℕ} = {G_{i} : i ∈ ℕ}. For each n ∈ ℕ, put G_{n} = {P_{α} : α ∈ Λ_{n}} and endow Λ_{n} with the discrete topology. 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}. It follows [20, Lemma 13] that f is a compact-covering and compact map. On the other hand, we have

Claim 1. M is locally separable.

Let a = (α_{i}) ∈ M. Then, {P_{α_{i}} } is a network at some point x_{a} ∈ X, and x_{a} ∈ P for some P ∈ P_{x_{a}} . Thus, there exists m ∈ ℕ such that P ∈ P_{m}. Hence, there exists a unique α ∈ Φ_{m} such that P ∈ Q_{m,α} . Therefore, P = P_{m,α,n} ∈ H_{m,n} for some n ∈ ℕ. Since P ∈ P_{x_{a}} ∩ H_{m,n} , x_{a} ∉ A_{m,n} . Noting that P ∩ P_{m,α,n} = ∅ for all n ≠ m. This implies that st(x, H_{m,n} ) = P. Then, H_{m,n} = G_{i_{0}} for some i_{0} ∈ ℕ and P = P_{α_{i_{0}}} . Thus, 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} is locally finite 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.

Claim 2. f is an mssc-map.

Let x ∈ X. For each n ∈ ℕ, since G_{n} is locally finite, there is an open neighborhood V_{n} of x such that V_{n} intersects at most finite members of G_{n}. Put

Θ_{n} = {α ∈ Λ_{n} : P_{α} ∩ V_{n} ≠ ∅}.

Then, Θ_{n} is finite and p_{n}f^{−1} (V_{n}) ⊂ Θ_{n}. Hence, \overline{p_{n}f^{−1} (V_{n})} is a compact subset of Λ_{n}, so f is an mssc-map.

(3) ⇒ (4) ⇒ (5) ⇒ (6). It is obvious.

(6) ⇒ (1). Let f : M → X be a sequentially-quotient π and mssc-map, where M be a locally separable metric space. By [9, Corollary 2.9], X has a point-star network {U_{n}}, where each U_{n} is a cs^{∗}-cover. For each n ∈ ℕ, put G_{n} = ⋀ _{i⩽n} U_{i}. Now, for each x ∈ X, let G_{x} = {st(x, G_{n}) : n ∈ ℕ}. Since each U_{n} is a cs^{∗}-cover, it implies that ⋃ {G_{x} : x ∈ X} is an sn-network for X. Hence, X is an sn-first countable space. On the other hand, since f is a sequentially-quotient mssc-map, it follows from Lemma 2.1 that X has a σ-locally finite Lindelöf cs-network P. We can assume that each P is closed under finite intersections. Then, each element of P is a cosmic subspace. By [19, Theorem 3.4], X has an so-cover consisting of ℵ_{0}-subspaces. Now, we only need to prove that X is an sn-metrizable space. In fact, since X is sn-first countable, X has an sn-network Q = ⋃ {Q_{x} : x ∈ X} with each Q_{x} = {Q_{n}(x) : n ∈ ℕ} is a countable sn-network at x. For each x ∈ X, put P_{x} = {P ∈ P : Q_{n}(x) ⊂ P for some n ∈ ℕ}. By using proof of [26, Lemma 7], we obtain P_{x} is an sn-network at x. Then, G = ⋃ {P_{x} : x ∈ X} is an sn-network for X. Since G ⊂ P, it implies that G is σ-locally finite. Thus, X is an sn-metrizable space. □

By Theorem 2.1 and [28, Lemma 2.7(2)], we have

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

(1) X has a locally countable weak base;

(2) X is a local ℵ_{0}-subspace and g-metrizable space;

(3) X has a σ-locally finite Lindelöf weak base;

(4) X is a compact-covering quotient compact and mssc-image of a locally separable metric space;

(5) X is a pseudo-sequence-covering quotient compact and mssc-image of a locally separable metric space;

(6) X is a subsequence-covering quotient compact and mssc-image of a locally separable metric space;

(7) X is a quotient π and mssc-image of a locally separable metric space.

Example 2.1. Let C_{n} be a convergent sequence containing its limit point p_{n} for each n ∈ ℕ, where C_{m} ∩ C_{n} = ∅ if m ≠ n. Let ℚ = {q_{n} : n ∈ ℕ} be the set of all rational numbers of the real line ℝ. Put M = ( ⨁ {C_{n} : n ∈ ℕ}) ⊕ ℝ and let X be the quotient space obtained from M by identifying each p_{n} in C_{n} with q_{n} in ℝ. Then, by the proof of [12, Example 3.1], X has a countable weak base and X is not a sequence-covering quotient π-image of a metric space. Hence,

(1) A space with a σ-locally finite Lindelöf sn-network ⇏ a sequence-covering π and mssc-image of a locally separable metric space.

(2) A space with a σ-locally finite Lindelöf weak base ⇏ a sequence-covering quotient π and mssc-image of a locally separable metric space.

Example 2.2. Using [10, 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 π-image of a metric space. This shows that regular properties of X can not be omitted in Theorem 2.1 and Corollary 2.1.

Example 2.3. S_{ω} is a Fréchet and ℵ_{0}-space, but it is not first countable. Thus, S_{ω} has a σ-locally finite Lindelöf cs-network. It follows from [3, Theorem 2.1] that X is a sequence-covering mssc-image of a locally separable metric space. Furthermore, since S_{ω} is not first countable, it doesn’t have a point-countable sn-network. Hence,

(1) A space with a σ-locally finite Lindelöf cs-network ⇏ a sequentiallyquotient π and mssc-image of a locally separable metric space.

(2) A sequence-covering quotient mssc-image of a locally separable metric space ⇏ X has a σ-locally finite Lindelöf sn-network.

Example 2.4. Using [15, Example 2.7], it is easy to see that X is a compactcovering quotient and compact image of a locally compact metric space, but it does not have a point-countable cs-network. Thus, a compact-covering quotient and compact image of a locally separable metric space ⇏ X has a σ-locally finite Lindelöf sn-network.

Example 2.5. There exists a space X having a locally countable sn-network, which is not an ℵ-space (see [6, Example 2.19]). Then, X has a σ-locally countable Lindelöf sn-network. Therefore,

(1) A space with a locally countable sn-network ⇏ X has a σ-locally finite Lindelöf sn-network.

(2) A space with a σ-locally countable Lindelöf sn-network ⇏ X has a σ-locally finite Lindelöf sn-network.

Acknowledgment. The author would like to express his thanks to the referee for his/her helpful comments and valuable suggestions.

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Department of Mathematics (Received 15 03 2011) Da Nang University of Education (Revised 26 07 2012) 459 Ton Duc Thang Lien Chieu Da Nang city Vietnam

    luongtuyench12@yahoo.com