Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 26, 2007, 33–49

\begin{matrix}c \\ ◯ \end{matrix} Xun Ge

Xun Ge

SPACES WITH A LOCALLY COUNTABLE SN-NETWORK

(submitted by M. A. Malakhaltsev)

Abstract. In this paper, we discuss a class of spaces with a locally countable sn-network. We give some characterizations of this class and investigate variance and inverse invariance of this class under certain mappings.

1. Introduction

sn-networks is a class of important networks between weak-bases and cs-networks. In past years, spaces with a locally countable weak-base and spaces with a locally countable cs-network had been investigated and many interesting results had been obtained ([17, 21, 22, 23, 24, 30, 31, 32, 33]). In this paper, we will discuss spaces with a locally countable sn-network. We give some characterizations of this class and investigate variance and inverse invariance of this class under certain mappings.

Throughout this paper, all spaces are assumed to be regular T_{1} and all mappings are continuous and onto. ℕ, ω and ω_{1} denote the set of all natural numbers, the first infinite ordinal and the first uncountable ordinal respectively. For a set D, |D| denotes the cardinal of D. {x_{n}} denotes a sequence, where the n-th term is x_{n}. Let X be a space and P ⊂ X. A sequence {x_{n}} converging to x in X is eventually in P if {x_{n} : n > k} ⋃{x} ⊂ P for some k ∈ ℕ; is frequently in P if {x_{n_{k}}} is eventually in P for some subsequence {x_{n_{k}}} of {x_{n}}. Let P be a family of subsets of X and f be a mapping defined on X. Then f(P) = {f(P) : P ∈ P}, ⋃ P and ⋂ P denote the union ⋃{P : P ∈ P} and the intersection ⋂{P : P ∈ P} respectively. Let P_{x} be a family of subsets of X. P_{x} is a network at x in X, if x ∈ ⋂ P_{x} and for each neighborhood U of x in X there is P ∈ P_{x} such that P ⊂ U. We refer the reader to [7] for notations and terminology not explicitly given here.

Key words and phrases. sn-network, cs-network, weak-base, perfectmapping, (strongly) Lindelöf mapping, finite subsequence-covering mapping.

2. Spaces with a Locally Countable sn-Network

Definition 2.1. [8, 9]. Let X be a space and let x ∈ X.

(1) P ⊂ X is called a sequential neighborhood of x if each sequence {x_{n}} converging to x is eventually in P.

(2) A subset U of X is called sequentially open if U is a sequential neighborhood of each of its points; a subset F of X is called sequentially closed if X − F is sequentially open.

(3) X is called a sequential space if each sequentially open subset of X is open in X, equivalently, if each sequentially closed subset of X is closed in X.

(4) X is called a k-space if for each A ⊂ X, A is closed in X if and only if A ⋂ K is closed in K for each compact subset K of X.

Remark 2.2. (1) P is a sequential neighborhood of x if and only if each sequence {x_{n}} converging to x is frequently in P.

(2) The intersection of finite sequential neighborhoods of x is a sequential neighborhood of x.

(3) the sequential spaces are the k-spaces.

Definition 2.3. Let P be a cover of a space X.

(1) P is called a k-network of X [27], if whenever K ⊂ U with K compact in X and U open in X, there is a finite F ⊂ P such that K ⊂ ⋃ F ⊂ U.

(2) P is called a cs^{∗}-network of X [10], if each convergent sequence S converging to a point x ∈ U with U open in X, then S is frequently in P ⊂ U for some P ∈ P.

(3) P is called a cs-network of X [15], if each convergent sequence S converging to a point x ∈ U with U open in X, then S is eventually in P ⊂ U for some P ∈ P.

Definition 2.4. Let P = ⋃{P_{x} : x ∈ X} be a cover of a space X, such that for each x ∈ X, the following conditions (a) and (b) are satisfied.

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

(b) If U, V ∈ P_{x}, then W ⊂ U ⋂ V for some W ∈ P_{x}.

SPACES WITH A LOCALLY COUNTABLE SN-NETWORK 35

(1) P is called a weak-base of X [1], if for G ⊂ X, G is open in X if and only if for each x ∈ G there is P ∈ P_{x} such that P ⊂ G, where P_{x} is called a weak neighborhood base at x in X.

(2) P is called an sn-network of X [11], if each element of P_{x} is a sequential neighborhood of x for each x ∈ X, where P_{x} is called an sn-network at x in X.

Remark 2.5. [23]. (1) weak-bases ⟹ sn-networks ⟹ cs-networks ⟹ cs^{∗}-networks.

(2) In a sequential space, weak-bases ⟺ sn-networks.

(3) sn-networks are called universal cs-networks in [20].

Example 2.6. In a k-space, sn-networks≠⇒ weak-bases.

Proof. Let X be the Stone-C̆ech compactification βℕ of ℕ. Then X is compact, and so it is a k-space. Since each convergent sequence in βℕ is trivial, P = {{x} : x ∈ X} is an sn-network of X. It is clear that P is not a weak-base of X. □

Definition 2.7. (1) A space X is called g-metrizable [8] (respectively snmetrizable [12], ℵ [27]) if X has a σ-locally finite weak-base (respectively sn-network, k-network).

(2) A space X is called g-second countable [29] (respectively sn-second countable [13], ℵ_{0} [26]) if X has a countable weak-base (respectively snnetwork, k-network).

(3) A space X is called g-first countable [1] (respectively sn-first countable [12]), if there is a countable weak neighborhood base (respectively sn-network) at x in X for each x ∈ X.

Remark 2.8. (1) g-first countable ⟺ sequential and sn-first countable.

(2) If X has a point countable weak-base (respectively sn-network), then X is g-first countable (respectively sn-first countable). So each g-metrizable (respectively sn-metrizable) space is g-first countable (respectively sn-first countable).

(3) g-metrizable (respectively g-second countable) ⟺ k- and snmetrizable (respectively k- and sn-second countable).

(4) X is an ℵ_{0}-space ⟺ X has a countable cs-network ⟺ X has a countable cs^{∗}-network.

(5) sn-first countable is called universally csf-countable in [20].

The following lemma is obtained by combining [19, Theorem 2.8.6] and [22, Corollary 5.1.13].

Lemma 2.9. The following are equivalent for a space X.

(1) X has a locally countable cs-network.

(2) X has a locally countable cs^{∗}-network.

(3) X has a locally countable k-network.

Now we give some set-theoretical axioms.

Set-Theoretical Axioms 2.10.

(1) CH (Continuum Hypothesis): 2^{ω} = ω_{1}.

(2) MA (Martin’s Axiom): Let k be a cardinal.

(i) A space X is called k^{+}-Baire if for each family {G_{α} : α ∈ A} consisting of open dense subsets of X, ⋂{G_{α} : α ∈ A} ≠ ∅, where |A| < k^{+}.

(ii) A space X is called ccc if each disjoint family consisting of open subsets of X is countable.

(iii) MA(k): Each compact, ccc space is k^{+}-Baire. (iv) MA: For each k, MA(k) holds, where ω < k < 2^{ω}.

(3) T OP (Thinning-out Principle): Let (P, ≤) be a partial ordered set.

(i) Q ⊂ P is called centered if whenever finite q_{1}, q_{2}, …, q_{n} ∈ Q, there is p ∈ P such that p ≤ q_{i} for all i = 1, 2, …, n.

(ii) A family {B_{α} : α < ω_{1}} is called cofinally centered on a set A if for each uncountable C ⊂ A there is an α < ω_{1}, such that {B_{β} ⋂ C : β ≥ α} is centered.

(iii) T OP: If Z, B are uncountable subsets of ω_{1} and {S_{α} : α ∈ B} is a family cofinally centered on Z with each S_{α} ⊂ α, then there is an uncountable Y ⊂ Z such that (Y ⋂ α) − S_{α} is finite for all α ∈ B.

Recall a space is an S-space if it is hereditarily separable and not hereditarily Lindelöf.

Lemma 2.11. [28, Theorem 7.2.3]. Under MA + ¬CH + T OP, there are no S-spaces.

Theorem 2.12. Under MA+¬CH +T OP, the following are equivalent for a space X.

(1) X has a locally countable sn-network.

(2) X is an sn-first countable space with a locally countable cs-network (respectively k-network, cs^{∗}-network).

(3) X is a locally sn-second countable space with a σ-locally countable sn-network

(4) X is a locally ℵ_{0}-space with a σ-locally countable sn-network.

(5) X is a locally hereditarily separable space with a σ-locally countable sn-network.

(6) X is a locally (hereditarily) Lindelöf space with a σ-locally countable sn-network.

Proof. (1) ⟹ (2). Note that a space with a locally countable sn-network is sn-first countable. So (1) ⟹ (2) by Remark 2.5(1) and Lemma 2.9.

(2) ⟹ (1). By Lemma 2.9, let P be a locally countable cs-network of X. We can assume that P is closed under finite intersections. For each x ∈ X, let {B_{n}(x) : n ∈ ℕ} be a countable sn-network at x in X, and let P_{x} = {P ∈ P : B_{n}(x) ⊂ P for some n ∈ ℕ}. Then each element of P_{x} is a sequential neighborhood of X. Put P′ = ⋃{P_{x} : x ∈ X}, then P′ ⊂ P is locally countable. It suffices to prove that P_{x} is a network at x in X for each x ∈ X. If not, there is an open neighborhood U of x such that P ⊄ U for each P ∈ P_{x}. Let {P ∈ P : x ∈ P ⊂ U} = {P_{m}(x) : m ∈ ℕ}. Then B_{n}(x) ⊄ P_{m}(x) for each n, m ∈ ℕ. Choose x_{n,m} ∈ B_{n}(x) − P_{m}(x). For n ≥ m, let x_{n,m} = y_{k}, where k = m + n(n − 1)/2. Then the sequence {y_{k} : k ∈ ℕ} converges to x. Thus , there is m, i ∈ ℕ such that {y_{k} : k ≥ i} ⋃{x} ⊂ P_{m}(x) ⊂ U. Take j ≥ i with y_{j} = x_{n,m} for some n ≥ m. Then x_{n,m} ∈ P_{m}(x). This is a contradiction.

(1) ⟹ (3). Let P be a locally countable sn-network of X. For each x ∈ X, there is an open neighborhood U of x such that P_{U} = {P ⋂ U : P ∈ P} is countable. It is easy to prove that P_{U} is a countable snnetwork of subspace U. So U is an sn-second countable space. Thus X is a locally sn-second countable space.

(3) ⟹ (4) ⟹ (5). It is known that sn-second countable ⟹ ℵ_{0} ⟹ hereditarily separable. So (3) ⟹ (4) ⟹ (5).

(5) ⟹ (6). It suffices to prove that X is locally hereditarily Lindelöf. Let x ∈ X and U be a hereditarily separable neighborhood of x. By Lemma 2.11, U is hereditarily Lindelöf. So X is locally hereditarily Lindelöf.

(6) ⟹ (1). Let P = ⋃{P_{n} : n ∈ ℕ} be a σ-locally countable snnetwork of a Locally Lindelöf space X, where each P_{n} is locally countable in X. Let x ∈ X and let U be a Lindelöf neighborhood of x. Let n ∈ ℕ. For each y ∈ U, there is an open neighborhood U_{y} of y such that U_{y} intersects at most countable many elements of P_{n}. The open cover {U_{y} : y ∈ U} of U has countable subcover V. Put V = ⋃ V, then U ⊂ V and V intersects at most countable many elements of P_{n}. So U intersects at most countable many elements of P_{n}. Moreover, U intersects at most countable many elements of P. Thus P is a locally countable sn-network of X. □

Remark 2.13. In Theorem 2.12, (1) ⟺ (2) ⟺ (3) ⟺ (4) ⟺ (6) without requiring MA + ¬CH + T OP involved. The reasons are as follows.

(a) MA + ¬CH + T OP are used only in the proof of (5) ⟹ (6).

(b) Because each ℵ_{0}-space is hereditarily Lindelöf, (4) ⟹ (6) without requiring MA + ¬CH + T OP involved.

It is natural to ask whether “MA + ¬CH + T OP ” in Theorem 2.12 can be omitted. The following Theorem 2.16 shows that the answer is “yes” if X is a k-space.

Lemma 2.14. [14, 22]. The following hold for a space X.

(1) If X is a compact space with a point countable k-network, then X is metrizable.

(2) If X is a k-space with a point countable k-network, then X is sequential.

(3) If X has a point countable cs^{∗}-network and each compact subset of X is metrizable, then X has a point countable k-network.

Lemma 2.15. If X is a k-space with a σ-locally countable cs^{∗}-network, then X is sequential.

Proof. Let P be a σ-locally countable cs^{∗}-network of X. Whenever K is a compact subset of X, put P_{K} = {P ⋂ K : P ∈ P}, then P_{K} is a σ-locally countable cs^{∗}-network of K. It is easy to see that P_{K} is a countable cs^{∗}- network of K, and so K has a countable k-network by Remark 2.8(4). By Lemma 2.14(1), K is metrizable. So X has a point-countable k-network by Remark 2.14(3), thus X is sequential by Remark 2.14(2). □

Theorem 2.16. The following are equivalent for a k-space X. (1) X has a locally countable sn-network.

(2) X is a topological sum of sn-second countable spaces.

(3) X is an sn-metrizable, locally (hereditarily) separable space.

(4) X is a locally (hereditarily) separable space with a σ-locally countable sn-network.

Proof. (1) ⟹ (2). X is a k-space with a locally countable cs-network, so X is a topological sum of ℵ_{0}-spaces([17, Theorem 1]). It is easy to see that sn-first countability is hereditary to subspace. Note that each sn-first countable, ℵ_{0}-space is sn-second countable([13, Theorem 2.1]). So X is a topological sum of sn-second countable spaces.

(2) ⟹ (3). Let X = ⊕{X_{α} : α ∈ Λ}, where each X_{α} is sn-second countable. Note that each X_{α} is a (hereditarily) separable, open subspace of X, So X is locally (hereditarily) separable. For each α ∈ Λ, let {P_{α,n} : n ∈ ℕ} be a countable sn-network of X_{α}. Put P_{n} = {P_{α,n} : α ∈ Λ} for each n ∈ ℕ, and put P = ⋃{P_{n} : n ∈ ℕ}, then P is a σ-locally finite sn-network of X. So X is an sn-metrizable space.

(3) ⟹ (4). It is clear.

(4) ⟹ (1). By Remark 2.13, it suffices to prove that X is locally Lindelöf. Let P be a σ-locally countable sn-network of X. X is a sequential space by Lemma 2.15, so P is a σ-locally countable k-network of X([30, Corollary 1.5]). Recalled a space is meta-Lindelöf if each open cover of it has a point countable open refinement. Thus X is hereditarily meta-Lindelöf([17, Proposition 1]). Each hereditarily meta-Lindelöf, locally separable space is locally Lindelöf([14, Proposition 8.7]), so X is locally Lindelöf. □

C. Liu and M. Dai proved that a space X has a locally countable weakbase if and only if it is a topological sum of g-second countable spaces [24, Theorem 2.1]. Combining Remark 2.8(3), we have the following corollary.

Corollary 2.17. A space X is a k-space with a locally countable snnetwork if and only if X has a locally countable weak-base.

The following examples to shows that “k” in Theorem 2.16 can not be omitted.

Example 2.18. There is a space with a locally countable sn-network, which is not a topological sum of ℵ_{0}-spaces.

Proof. Let D is a discrete space, where |D| = 2^{ω}. By [3, Example 4.2], there is an almost disjoint family {P_{α} : α < 2^{ω}} consisting of countable infinite subsets of D such that for each uncountable subset P of D, there is α < 2^{ω} such that P_{α} ⊂ P. Let {P_{α,n} : n ∈ ℕ} be a mutually disjoint family consisting of infinite subsets of P_{α}. For each α < 2^{ω} and each n ∈ ℕ, choose p_{α,n} ∈ \overline{P_{α,n}} − P_{α,n}, where \overline{P_{α,n}} is the closure of P_{α,n} in the Stone-C̆ech compactification βD of D. Put X = D ⋃{p_{α,n} : α < 2^{ω}, n ∈ ℕ}, and X is endowed the subspace topology of βD.

Claim 1. X has a σ-locally countable sn-network.

By [22, Example 5.1.18(1)], X has a σ-locally countable cs-network P. Note that each compact subset of X is finite [22, Example 1.5.5], so each convergent sequence of X is finite. Thus we can assume that P is closed under finite intersections. It is easy to see that P is an sn-network of X. So X has a σ-locally countable sn-network.

Claim 2. X is not a topological sum of ℵ_{0}-spaces [22, Example 5.1.18(1)]. □

Example 2.19. There is a space with a locally countable sn-network, which is not an ℵ-spaces.

Proof. Let X = ω_{1} ⋃(ω_{1} × {1/n : n ∈ ℕ}). Define a neighborhood base B_{x} for each x ∈ X for the desired topology on X as follows.

(1) If x ∈ X − ω_{1}, then B_{x} = {{x}}.

(2) If x ∈ ω_{1}, then B_{x} = {{x} ⋃(⋃{V (n, x) × {1/n} : n ≥ m}) : m ∈ ℕ and V (n, x) is a neighborhood of x in ω_{1} with the order topology}.

By [17, Example 1], X has a locally countable k-network, which is not an ℵ-space. It suffices to prove that X is sn-first countable by Remark 2.13.

Let x ∈ X. If x ∈ X −ω_{1}, then {{x}} is a countable sn-network at x in X. If x ∈ ω_{1}, put P_{x} = {P_{x,m} : m ∈ ℕ}, where P_{x,m} = {x} ⋃{(x, 1/n) : n ≥ m}. Then P_{x} is a countable network at x in X. We only need to prove that each P_{x,m} is a sequential neighborhood of x.

Let {x_{n}} be a sequence converging to x. Put K = {x} ⋃{x_{n} : n ∈ ℕ}, then K is a compact subset of X. By [17, Example 1], we have the following facts.

Fact 1. K ⋂ ω_{1} is finite.

Fact 2. K − ⋃{{y} ⋃{(y, 1/n) : n ∈ ℕ} : y ∈ K ⋂ ω_{1}} is finite.

Case 1. If there is y ∈ K ⋂ ω_{1} such that y = x_{n} for infinite many n ∈ ℕ, i.e., there is a subsequence {x_{n_{k}}} of {x_{n}} such that y = x_{n_{k}} for each k ∈ ℕ, then y = x, So {x_{n}} is frequently in P_{x,m}.

Case 2. If Case 1 does not hold, without loss of the generalization, we may assume K ⋂ ω_{1} = {x} by Fact 1. By Fact 2, K − {x} ⋃{(x, 1/n) : n ∈ ℕ} is finite. If there is y ∈ K − {x} ⋃{(x, 1/n) : n ∈ ℕ} such that y = x_{n} for infinite many n ∈ ℕ, then {x_{n}} is frequently in P_{x,m} by a similar way in the proof of Case 1. Conversely, there is k_{0} ∈ ℕ such that {x} ⋃{x_{n} : n ≥ k_{0}} ⊂ {x} ⋃{(x, 1/n) : n ∈ ℕ}. So {x_{n}} is eventually in P_{x,m}.

By the above Case 1 and Case 2, P_{x,m} is a sequential neighborhood of x by Remark 2.2(1). □

Recalled a space X is sequentially separable [6] if X has a countable subset D such that for each x ∈ X, there is a sequence {x_{n}} in D converging to x, where D is a sequentially dense subset of X. It is know that each sequentially separable space is separable.

Proposition 2.20. Let X have a point countable sn-network P. If X is sequentially separable, then P is countable. So X is sn-second countable.

Proof. Let D be a sequentially dense subset of X, and let P = {P_{x} : x ∈ X}, where P_{x} is an sn-network at x in X for each x ∈ X. For each x ∈ D, (P)_{x} is countable because P is point countable, where (P)_{x} = {P ∈ P : x ∈ P }. Furthermore, ⋃{(P)_{x} : x ∈ D} is countable. For each x ∈ X and P ∈ P_{x}, there is a sequence S in D converging to x. Note that P is a sequential neighborhood of x. S is eventually in P, and so P ⋂ D ≠ ∅. This proves that each element of P intersects with D, thus P = ⋃{(P)_{x} : x ∈ D}. So P is countable. □

Corollary 2.21. Let X have a σ-locally countable (or point countable) sn-network P. If X is locally sequentially separable, then P is locally countable in X. So X has a locally countable sn-network.

Proof. Since σ-locally countable ⟹ point countable, we only need to prove parenthetic part.

Let X be locally sequentially separable. For each x ∈ X, there is an open neighborhood U of x such that U is sequentially separable. It is clear that {P ⋂ U : P ∈ P} is a point countable sn-network of U. {P ⋂ U : P ∈ P} is countable by Proposition 2.20, So P is locally countable in X. □

The following example shows that “sequentially separable” in Proposition 2.20 can not be relaxed to “separable”, which is due to [16, Example 1].

Example 2.22. There is a separable, sn-metrizable space, which is not an ℵ_{0}-spaces.

Proof. Let ℚ ⊂ X ⊂ ℝ and |X| > ω, where ℚ and ℝ are the set of all rational numbers and the set of all real numbers respectively. Let Y = X ⋃(⋃{ℚ × {1/n} : n ∈ ℕ}). Define a neighborhood base B_{y} for each y ∈ Y for the desired topology on Y as follows.

(1) If y ∈ Y − X, then B_{y} = {{y}}.

(2) If y ∈ X, then B_{y} = {{y} ⋃(⋃{([a_{y,n}, y) ⋂ ℚ) × {1/m} : n ≥ m}) : m ∈ ℕ and y > a_{y,n} ∈ ℝ}.

Then Y is a separable, ℵ-space and not an ℵ_{0}-space [16, Example 1]. On the other hand, each compact subset of Y is finite [16, Example 1]. By a similar way as in the proof of Example 2.18(claim 1), we can prove Y has a σ-locally finite sn-network. That is, Y is an sn-metrizable space. □

3. Mappings on Spaces with a Locally Countable sn-Network

In this section, we discuss invariance and inverse invariance of spaces with a locally countable sn-network under certain mappings

Definition 3.1. Let f : X ⟶ Y be a mapping.

(1) f is called a perfect mapping [7] if f is closed and f^{−1}(y) is a compact subset of X for each y ∈ Y.

(2) f is called a Lindelöf mapping [31] (respectively strongly Lindelöf mapping [31]) if for each y ∈ Y, f^{−1}(y) is a Lindelöf subset of X (respectively f^{−1}(\overline{U}) is a Lindelöf subset of X for some neighborhood U of y in Y).

(3) f is called a 1-sequence-covering mapping [23] if for each y ∈ Y there is x ∈ f^{−1}(y), such that whenever {y_{n}} is a sequence converging to y in Y, there is a sequence {x_{n}} converging to x in X with each x_{n} ∈ f^{−1}(y_{n}).

(4) f is called a finite subsequence-covering mapping [25] if for each y ∈ Y there is a finite subset F of f^{−1}(y), such that for any sequence S in Y converging to y, there is a sequence L in X converging to some x ∈ F and f(L) is a subsequence of S.

(5) f is a sequentially-quotient mapping [4] if whenever S is a convergent sequence in Y there is a convergent sequence L in X such that f(L) is a subsequence of S.

(6) f is a quotient mapping [7] if whenever U ⊂ Y, f^{−1}(U) is open in X if and only if U is open Y.

We call a space X to be point-G_{δ} if for each x ∈ X, there is a sequence {U_{n}} of neighborhoods of x in X such that {x} = ⋂{U_{n} : n ∈ ℕ}. It is known that if a space X has a locally countable cs-network, then X is point-G_{δ} [26, (D)].

Remark 3.2. [19]. (1) 1-sequence-covering mappings (sequentially-quotient and finite-to-one mappings, respectively) ⟹ finite subsequence-covering mappings ⟹ sequentially-quotient mappings.

(2) Closed mappings ⟹ quotient mappings.

(3) If the domain is point-G_{δ}, then closed mappings ⟹ sequentiallyquotient mappings

(4) If the domain is sequential, then quotient mappings ⟹ sequentially-quotient mappings.

(5) Quotient mappings preserve k-spaces and perfect mappings inversely preserve k-spaces.

Definition 3.3. [20]. Let X be a space. Put

σ = {P ⊂ X : P is sequentially open in X}.

The (X, σ), the set X with the topology σ, is called the sequential coreflection of X, which is denoted by σX.

Definition 3.4. [2]. Let T_{0} = {a_{n} : n ∈ ℕ} be a sequence converging to x_{0} ∉ T_{0}, and let T_{n} be a sequence converging to a_{n} ∉ T_{n} for each n ∈ ℕ. Let T be the topological sum of {T_{n} ⋃{a_{n}} : n ∈ ℕ}. S_{ω} is defined as a quotient space obtained from T by identifying all point a_{n} ∈ T to the point x_{0}.

The following lemma is obtained by combining [20, Theorem 3.6] and [20, Theorem 3.13].

Lemma 3.5. [20]. Let X be a point-G_{δ} space and contain no closed subspace having S_{ω} as its sequential coreflection. If X has a point-countable cs-network, then X is sn-first countable.

Lemma 3.6. [21]. Let f : X ⟶ Y be a perfect mapping and X have a G_{δ}-diagonal. If Y has a locally countable k-network, then X has a locally countable k-network.

Lemma 3.7. [11]. Let f : X ⟶ Y be a closed mapping and X be point-G_{δ}. If F is sequentially closed in X, then f(F) is sequentially closed in Y.

Theorem 3.8. Let f : X ⟶ Y be a perfect mapping and X have a G_{δ}- diagonal. If Y has a locally countable sn-network, then X has a locally countable sn-network.

Proof. If Y has a locally countable sn-network, then X has a locally countable cs-network by Remark 2.5(1), Lemma 2.9 and Lemma 3.6. We only need to prove that X is sn-first countable by Remark 2.13. Since X has a G_{δ}-diagonal, X is point-G_{δ}. By Lemma 3.5, It suffices to prove that X contains no closed subspace having S_{ω} as its sequential coreflection.

Assume X contains closed subspace S having S_{ω} as its sequential coreflection. Put g = f|_{σs} : σS ⟶ σf(S).

Claim 1. g is closed.

Proof. Let A be a closed subset of σS, then A is sequentially closed in S. It is clear that f : S ⟶ f(S) is closed and S is point-G_{δ}. So f(A) is sequentially closed in f(S) by Lemma 3.7, thus f(A) is closed in σf(S).

Claim 2. g^{−1}(y) is compact in σS for each y ∈ σf(S).

Proof. Let y ∈ σf(S). Note that X has a G_{δ}-diagonal and f^{−1}(y) is compact in X, so f^{−1}(y) is metrizable [5]. Therefore, the topology on the sequential coreflection of f^{−1}(y) ⋂ S is equivalent to the induced topology of subspace S of X. Thus g^{−1}(y) = f^{−1}(y) ⋂ S is compact in σS.

By the above two claims, g is perfect. Since S_{ω}, which is homeomorphic to σS, is a F réchet, ℵ-space and perfect mappings preserve F réchet, ℵspaces, σf(S) is a F réchet, ℵ-space. On the other hand, Y is sn-first countable, so f(S), as a subspace of Y, is sn-first countable. By [20, Theorem 3.13], σf(S) is g-first countable, so σf(S) is metrizable [11, Theorem 2.4], and so σS is metrizable [5]. This contradicts that S_{ω} is not metrizable. □

We have the following corollary by Corollary 2.17, Remark 3.2(5) and Theorem 3.8.

Corollary 3.9. Let f : X ⟶ Y be a perfect mapping and X have a G_{δ}-diagonal. If Y has a locally countable weak-base, then X has a locally countable weak-base.

Example 3.10. A perfect image of a g-second countable space has not any locally countable sn-network.

Proof. Let X = {0} ⋃ ℕ ⋃(ℕ × ℕ), F = {F ⊂ ℕ : F is finite}, ℕ^{ℕ} = {f : fis a correspondence from ℕ to ℕ}. For n, m, k ∈ ℕ, F ∈ F, and f ∈ ℕ^{ℕ}, put V (n, m) = {n} ⋃{(n, k) : k ≥ m}, H(F, f) = ⋃{V (n, f(n)) : n ∈ ℕ − F }. Define a neighborhood base B_{x} for each x ∈ X for the desired topology on X as follows.

(1) If x ∈ ℕ × ℕ, then B_{x} = {{x}}.

(2) If x ∈ ℕ, then B_{x} = {V (x, m) : m ∈ ℕ}.

(3) If x = 0, then B_{x} = {{x} ⋃ H(F, f) : F ∈ F, f ∈ ℕ^{ℕ}}.

Let Y be the quotient space obtained from X by shrinking the set {0} ⋃ ℕ to a point, f : X ⟶ Y be a natural mapping. Then

Claim 1. f is perfect and X is g-second countable [18, Example 3.1]. Claim 2. Y is not sn-first countable [11, Example 3.2], so Y has not

any locally countable sn-network. □

Which mappings preserve spaces with a locally countable sn-network? We give some answers for this question.

Lemma 3.11. Let f : X ⟶ Y be a finite subsequence-covering mapping. If X is sn-first countable, then Y is sn-first countable.

Proof. Let y ∈ Y. Then there is a finite subset F of f^{−1}(y), such that for any sequence S in Y converging to y, there is a sequence L in X converging to some x ∈ F and f(L) is a subsequence of S. X is sn-first countable, for each x ∈ F, let P_{x} = {P_{x,n} : n ∈ ℕ} be a decreasing sn-network at x in X. Put F_{y} = {⋃{f(P_{x,n}) : x ∈ F } : n ∈ ℕ}. Then F_{y} is countable decreasing.

(1) F_{y} is a network at y in Y. In fact, let U be an open neighborhood of y, then F ⊂ f^{−1}(y) ⊂ f^{−1}(U). For each x ∈ F, there is n_{x} ∈ ℕ such that x ∈ P_{x,n_{x}} ⊂ f^{−1}(U), so y ∈ f(P_{x,n_{x}}) ⊂ U. Put n_{0} = max{n_{x} : x ∈ F }, then P_{x,n_{0}} ⊂ P_{x,n_{x}} for each x ∈ F. So y ∈ ⋃{f(P_{x,n_{0}}) : x ∈ F } ⊂ ⋃{f(P_{x,n_{x}}) : x ∈ F } ⊂ U.

(2) Let ⋃{f(P_{x,n_{1}}) : x ∈ F }, ⋃{f(P_{x,n_{2}}) : x ∈ F } ∈ F_{y}. Put n_{0} = max{n_{1}, n_{2}}, then ⋃{f(P_{x,n_{0}}) : x ∈ F } ∈ F_{y} and ⋃{f(P_{x,n_{0}}) : x ∈ F } ⊂ (⋃{f(P_{x,n_{1}}) : x ∈ F }) ⋂(⋃{f(P_{x,n_{2}}) : x ∈ F }).

(3) ⋃{f(P_{x,n}) : x ∈ F } is a sequential neighborhood of y for each n ∈ ℕ. In fact, let S be a sequence in Y converging to y. Then there is a sequence L in X converging to some x_{0} ∈ F and f(L) is a subsequence of S. For each n ∈ ℕ. Since P_{x_{0},n} is a sequential neighborhood of x, L is eventually in P_{x_{0},n}. So f(L) is eventually in f(P_{x_{0},n}), hence S is frequently in f(P_{x_{0},n}). Moreover, S is frequently in ⋃{f(P_{x,n}) : x ∈ F }. By Remark 2.2(1), ⋃{f(P_{x,n}) : x ∈ F } is a sequential neighborhood of y. □

Lemma 3.12. Let f : X ⟶ Y be a closed, Lindelöf mapping. If P is a locally countable family of subsets of X, then f(P) is a locally countable family of subsets of Y.

Proof. Let P = {P_{α} : α ∈ Λ} be a locally countable family of subsets of X and let y ∈ Y. For each x ∈ f^{−1}(y), there is an open neighborhood U_{x} of x such that {α ∈ Λ : U_{x} ⋂ P_{α} ≠ ∅} is countable. f^{−1}(y) ⊂ ⋃{U_{x} : x ∈ f^{−1}(y)} and f^{−1}(y) is Lindelöf, so there is a countable subset B of f^{−1}(y) such that f^{−1}(y) ⊂ ⋃{U_{x} : x ∈ B}. Put U = ⋃{U_{x} : x ∈ B}. It is clear that {α ∈ Λ : U ⋂ P_{α} ≠ ∅} is countable. Note that f is closed. By [7, Theorem 1.4.13], there is an open neighborhood V of y such that f^{−1}(V) ⊂ U. Thus Λ′ = {α ∈ Λ : f^{−1}(V) ⋂ P_{α} ≠ ∅} is countable. It is easy to check that {α ∈ Λ : V ⋂ f(P_{α}) ≠ ∅} = Λ′. So {α ∈ Λ : V ⋂ f(P_{α}) ≠ ∅} is countable. This proves that f(P) is a locally countable family of subsets of Y. □

Theorem 3.13. Let f : X ⟶ Y be a closed, finite-to-one mapping. If X has a locally countable sn-network, then Y has a locally countable sn-network.

Proof. Let P be a locally countable sn-network of X. Then f is sequentially quotient by Remark 3.2(3), and so Y is sn-first countable by Remark 3.2(1) and Lemma 3.11. Since sequentially quotient mappings preserve cs^{∗}-networks [19, Proposition 2.7.3], f(P) is a cs^{∗}-network of Y. f(P) is locally countable by Lemma 3.12, so f(P) is a locally countable cs^{∗}-network of Y. Thus Y has a locally countable sn-network by Remark 2.13. □

Question 3.14. Do closed, countable-to-one mappings preserve spaces with a locally countable sn-network?

A clopen mapping means an open and closed mapping.

Theorem 3.15. Let f : X ⟶ Y be a clopen, Lindelöf mapping. If X has a locally countable sn-network, then Y has a locally countable sn-network.

Proof. Let P = ⋃{P_{x} : x ∈ X} be a locally countable sn-network of X. Since f is closed, Lindelöf, by a similar way as in the proof of Theorem 3.13, f(P) is a locally countable cs^{∗}-network of Y. It suffices to prove that Y is sn-first countable by Remark 2.13. Let y ∈ Y. Put F_{y} = {f(P) : P ∈ P_{x} and x ∈ f^{−1}(y)}, then F_{y} ⊂ f(P), so F_{y} is locally countable. Note that y ∈ ⋂ F_{y}, F_{y} is countable. It is clear that F_{y} is a network at y in Y. We only need to prove that each element of F_{y} is a sequential neighborhood of y. Let f(P) ∈ F_{y} and {y_{k}} be a sequence in Y converging to y. Then there is x ∈ f^{−1}(y) such that P ∈ P_{x}. Since X is point-G_{δ}, {x} = ⋂{U_{n} : n ∈ ℕ}, where each U_{n} is open in X and \overline{U_{n+1}} ⊂ U_{n}. For each n ∈ ℕ, y ∈ f(U_{n}) and f(U_{n}) is open as f is open, so there is m_{n} ∈ ℕ such that y_{k} ∈ f(U_{n}) for each k ≥ m_{n}. Pick x_{n} ∈ U_{n} such that f(x_{n}) = y_{m_{n}}. Since f is closed, it is not difficult to prove that the sequence {x_{n}} converges to x ∈ P. P is a sequential neighborhood of x, so {x_{n}} is eventually in P. Consequently, {f(x_{n})} is eventually in f(P), so {y_{k}} is frequently in f(P). By Remark 2.2(1), f(P) is a sequential neighborhood of y. □

Corollary 3.16. Let f : X ⟶ Y be an open, perfect mapping. If X has a locally countable sn-network, then Y has a locally countable sn-network.

Clopen mappings preserve spaces with a locally countable weak-base [24, Theorem 4.7]. But the following question is still open.

Question 3.17. Do clopen mappings preserve spaces with a locally countable sn-network (respectively cs-network)?

Lemma 3.18. Let f : X ⟶ Y be a strongly Lindelöf mapping. If P is a locally countable family of subsets of X, then f(P) is a locally countable family of subsets of Y.

Proof. Let P = {P_{α} : α ∈ Λ} be a locally countable family of subsets of X and let y ∈ Y. Then there is a neighborhood W of y in Y such that f^{−1}(\overline{W}) is a Lindelöf subset of X. It suffices to prove that \{α ∈ Λ : W ⋂ f(P_{α}) ≠ ∅\} is countable. For each x ∈ f^{−1}(\overline{W}), there is an open neighborhood U_{x} of x such that {α ∈ Λ : U_{x} ⋂ P_{α} ≠ ∅} is countable. f^{−1}(\overline{W}) ⊂ ⋃\{U_{x} : x ∈ f^{−1}(\overline{W})\} and f^{−1}(\overline{W}) is Lindelöf, so there is a countable subset B of f^{−1}(\overline{W}) such that f^{−1}(\overline{W}) ⊂ ⋃\{U_{x} : x ∈ B\}. It is easy to see that {α ∈ Λ : (⋃{U_{x} : x ∈ B}) ⋂ P_{α} ≠ ∅} is countable, so Λ′ = {α ∈ Λ : f^{−1}(W) ⋂ P_{α} ≠ ∅} is countable. It is easy to check that {α ∈ Λ : W ⋂ f(P_{α}) ≠ ∅} = Λ′. This completes the proof. □

Theorem 3.19. Let X have a locally countable sn-network. If one of the following holds, then Y has a locally countable sn-network.

(1) f is finite subsequence-covering, strongly Lindelöf.

(2) f is 1-sequence-covering, strongly Lindelöf.

(3) f is sequentially-quotient, finite-to-one, strongly Lindelöf.

Proof. We only need to prove part (1) by Remark 3.2(1). Let f : X ⟶ Y be a finite subsequence-covering, strongly Lindelöf mapping and P be a locally countable sn-network of X. Then Y is sn-first countable by lemma 3.11 and f(P) is a locally countable family of subsets of Y by Lemma 3.18. By a similar way as in the proof of Theorem 3.13, we can prove f(P) is a cs^{∗}-network of Y. So Y has a locally countable sn-network by Remark 2.13. □

The following corollary is obtained by Remark 3.2, Corollary 2.17, Theorem 3.13 and Theorem 3.19.

Corollary 3.20. Let X have a locally countable weak-base. If one of the following holds, then Y has a locally countable weak-base.

(1) f is closed, finite-to-one.

(2) f is finite subsequence-covering, quotient, strongly Lindelöf.

(3) f is 1-sequence-covering, quotient, strongly Lindelöf.

(4) f is quotient, finite-to-one, strongly Lindelöf.

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Department of Mathematics, Suzhou University, Suzhou, 215006, P.R.China E-mail address: zhugexun@163.com

Received May 12, 2007