MATEMATIQKI VESNIK

59 (2007), 15–21

UDK 515.122

originalni nauqni rad research paper

ON SEQUENCE-COVERING msss-MAPS

Zhaowen Li, Qingguo Li and Xiangnan Zhou

Abstract. This paper gives characterizations of metric spaces under some sequence-covering msss-maps by means of certain kind of σ-locally countable networks.

1. Introduction and definitions

A study of some images of metric spaces under certain maps is an important task on general topology. The paper [1] introduced the concept of msssmaps, and established the relationships between spaces with σ-locally countable k-networks (bases) and metric spaces by means of msss-maps. In this paper, we study some spaces with σ-locally countable networks, and give characterizations of some sequence-covering msss-images of metric spaces.

In this paper all spaces are regular and T_{1}, all maps are continuous and onto. N denotes the set of all natural numbers, ω denotes N ∪ {0}. For two family A and B of subsets of a space X, Denote A ∧ B = {A ∩ B : A ∈ A and B ∈ B}. For the usual product space ∏_{i∈N} X_{i}, p_{i} denotes the projection from ∏_{i∈N} X_{i} onto X_{i} For a space X and each x_{n} ∈ X, (x_{n}) denotes a point of the usual product space X^{ω} whose n-th coordinate is x_{n}.

Definition 1.1. [10] Let X be a space, and P ⊂ X. Then,

(1) A sequence {x_{n}} in X is eventually in P if {x_{n}} converges to x, and there exists m ∈ N such that {x} ∪ {x_{n} : n ≥ m} ⊂ P;

(2) P is a sequential neighborhood of x in X if whenever a sequence {x_{n}} in X converges to x, then {x_{n}} is eventually in P;

(3) P is sequentially open in X if P is a sequential neighborhood at each of its points;

(4) X is a sequential space if any sequentially open subset of X is open in X.

AMS Subject Classification: 54E99, 54C10; 54D55.

Keywords and phrases: msss-maps; 1-sequence-covering maps; sequence-covering maps; strong compact-covering maps; weak-bases; bases; s-networks; cs-networks.

The project is supported by the NSF of China (No. 10471035), the NSF of Hunan Province in China (No. 06JJ20046) and the NSF of Education Department of Hunan Province in China (No. 06C461).

Definition 1.2. Let P = ⋃{P_{x} : x ∈ X} be a family of subsets of a space X satisfying for each x ∈ X,

(1) P_{x} is a network of x in X. i.e., x ∈ ∩P_{x} and for x ∈ U with U open in X, P ⊂ U for some P ∈ P_{x},

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

P is a weak-base for X [8] if G ⊂ X such that for each x ∈ G, there exists P ∈ P_{x} satisfying P ⊂ G, then G is open in X. P is an sn-network (i.e., sequential neighborhood network) for X if each element of P_{x} is a sequential neighborhood of x in X. P is an so-network (i.e., sequential open network) for X if each element of P_{x} is sequentially open in X. The above P_{x} respectively is a weak-base, an sn-network and an so-network of x in X.

Definition 1.3. [4] For a space X, let P be a family of subsets of X, there exists P_{x} ⊂ P^{ω} holding the following property: if (P_{n}) ∈ P_{x}, then {P_{n} : n ∈ N} is a decrease network of x in X. Denote P ≈ ⋃{P_{x} : x ∈ X}.

(1) P is an s-network (i.e., sequential network) for X if, whenever P ⊂ X, and for x ∈ P and each (P_{n}) ∈ P_{x}, P_{m} ⊂ P for some m ∈ M, then P is sequentially open in X;

(2) P is a sequential quasi-bases for X if, whenever P ⊂ X, and for x ∈ P and each (P_{n}) ∈ P_{x}, P_{m} ⊂ P for some m ∈ M, then P is open in X;

(3) P is a Fréchet quasi-bases for X if, whenever P ⊂ X, and for x ∈ P and each (P_{n}) ∈ P_{x}, P_{m} ⊂ P for some m ∈ M, then P is a neighborhood of x in X.

Definition 1.4. Let P be a family of subsets of a space X.

(1) P is a cs-network [9] for X if whenever {x_{n}} is a sequence converging to a point x ∈ U with U open in X, then there are P ∈ P and m ∈ N such that {x_{n} : n ≥ m} ∪ {x} ⊂ P ⊂ U;

(2) P is a cs^{∗}-network for X if whenever {x_{n}} is a sequence converging to a point x ∈ U with U open in X, then there are a subsequence {x_{n_{i}}} and P ∈ P such that {x_{n_{i}} : i ∈ N} ∪ {x} ⊂ P ⊂ U.

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

(1) f is a msss-map [1] (i.e., metrizably stratified strong s-map) 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}} of open neighborhoods of y such that each p_{i}f^{−1}(V_{i}) is separable in X_{i};

(2) f is a 1-sequence-covering map [2] if for each y ∈ Y, there exists x ∈ f^{−1}(y) satisfying the following condition (∗): whenever y_{n} → y, then there exists x_{n} ∈ f^{−1}(y_{n}) such that x_{n} → x;

(3) f is a 2-sequence-covering map [2] if for each y ∈ Y and each x ∈ f^{−1}(y) satisfying the above condition (∗);

(4) f is a sequence-covering map [13] (resp. compact-covering map) if each convergent sequence (including its limit point) of Y (resp. each compact subset of Y) is the image of some compact subset of X;

On sequence-covering msss-maps 17

(5) f is a strong sequence-covering map [5] if each convergent sequence (including its limit point) in Y is the image of some convergent sequence (including its limit point) in X;

(6) f is a strong compact-covering map [5] if it is both strong sequence-covering and compact-covering;

(7) f is a sequentially quotient map [7] if whenever R ⊂ Y and f^{−1}(R) is sequentially open in X, then R is sequentially open in Y.

2. On 1-sequence-covering msss-images

Theorem 2.1. A space X is a 1-sequence-covering msss-image of a metric space if and only if X has a σ-locally countable sn-network.

Proof. Sufficiency. Suppose P is a σ-locally countable sn-network for X. Let P = ⋃{P_{i} : i ∈ N}, where each P_{i} = {P_{α} : α ∈ A_{i}} is locally countable in X. We can assume that P_{i} is closed under finite intersections and X ∈ P_{i} ⊂ P_{i+1}. For each i ∈ N, endow A_{i} with discrete topology; then A_{i} is a metric space. Put

M = {α = (α_{i}) ∈ ∏_{i∈N} A_{i} : {P_{α_{i}} : i ∈ N} is a network of some point x_{α} in X} ,

and endow M with the subspace topology induced from the product topology of a family {A_{i} : i ∈ N} of metric spaces, then M is a metric space. Since X is Hausdorff, x_{α} is unique in X for each α ∈ M. We define f : M → X by f(α) = x_{α} for each α ∈ M. Since P is a σ-locally countable sn-network for X, f is onto. For each α = (α_{i}) ∈ M , f(α) = x_{α}. Suppose V is an open neighborhood of x_{α} in X, there exists n ∈ N such that x_{α} ∈ P_{α_{n}} ⊂ V, set W = {c ∈ M : the n-the coordinate of c is α_{n}}, then W is an open neighborhood of α in M, and f(W) ⊂ P_{α_{n}} ⊂ V. Hence f is continuous. We will show that f is a 1-sequence-covering msss-map.

(i) f is an msss-map.

For each x ∈ X and each i ∈ N, there exists an open neighborhood V_{i} of x in X such that {α_{i} ∈ A_{i} : P_{α} ∩ V_{i} ≠ ∅} is countable. Put

B_{i} = {α_{i} ∈ A_{i} : P_{α} ∩ V_{i} ≠ ∅},

then p_{i}f^{−1}(V_{i}) ⊂ B_{i}. Thus p_{i}f^{−1}(V_{i}) is separable in A_{i}. Hence f is a msss-map. (ii) f is a 1-sequence-covering map. For each x ∈ X, by the definition of P, there exists (α_{i}) ∈ ∏_{i∈N} A_{i} such that

{P_{α_{i}} : i ∈ N} ⊂ P is an sn-network of x in X. Denote β = (α_{i}), then β ∈ f^{−1}(x). For each n ∈ N, let R_{n} = {(γ_{i}) ∈ M : if i ≤ n, then γ_{i} = α_{i}}. Then {R_{n} : n ∈ N} is a decreasing neighborhood base of β in M. For each n ∈ N, it is easy to check that f(R_{n}) = ⋂_{i≤n} P_{α_{i}}. Now suppose x_{j} → x in X. For each n ∈ N, since f(B_{n}) is a sequential neighborhood of x in X, there exists i(n) ∈ N such that if i ≥ i(n), then x_{i} ∈ f(R_{n}). Thus f^{−1}(x_{i}) ∩R_{n} ≠ ϕ. We may assume 1 < i(n) < i(n + 1). For each j ∈ N, let

β_{j} ∈ \{ \begin{matrix}f^{−1}(x_{j}), & if j < i(1), \\ f^{−1}(x_{j}) ∩ R_{n}, & if i(n) ≤ j < i(n + 1), n ∈ N. \end{matrix}

Then it is easy to show that sequence {β_{j}} converges to β in M. Hence f is a 1-sequence-covering map.

Necessity. Suppose f: M → X is a 1-sequence-covering msss-map, where M is a metric space. Since f is a msss-map, then there exists a base B for M such that P^{∗} = {f(B) : B ∈ B} is a σ-locally countable network for X by Lemma 1.2 of [1]. For each x ∈ X, β ∈ f^{−1}(x) satisfies the condition (∗) of Definition 1.5 (2). Put

P_{x} = {f(B) : β_{x} ∈ B ∈ B}, P = ⋃{P_{x} : x ∈ X}.

It is easy to show that each P_{x} is an sn-network of x in X, and P is an sn-network for X. Obviously, P ⊂ P^{∗}. Hence X has a σ-locally countable sn-network.

Corollary 2.2. A space X is a 1-sequence-covering and quotient msss-image of a metric space if and only if X has a σ-locally countable weak-base.

Proof. Sufficiency. Suppose X has a σ-locally countable weak-base, then X is a sequential space with a σ-locally countable sn-network by [3, Proposition 1.6.15, Corollary 1.6.18]. Thus X is a 1-sequence-covering msss-image of a metric space by Theorem 2.1. This 1-sequence-covering msss-map is quotient by Lemma 2.1 of [14].

Necessity. Suppose X is a 1-sequence-covering and quotient msss-image of a metric space. Then X is a sequential space with a σ-locally countable sn-network P. It is easy to prove that P is a σ-locally countable weak-base for X.

3. On 2-sequence-covering msss-images

The following Theorem 3.1 can be proved by Lemma 3.1 of [2] according to the proof of Theorem 2.1.

Theorem 3.1. A space X is a 2-sequence-covering msss-image of a metric space if X has a σ-locally countable so-network.

Corollary 3.2. The following are equivalent for a space X:

(1) X has a σ-locally countable base;

(2) X is a 2-sequence-covering and quotient msss-image of a metric space;

(3) X is an open msss-image of a space having a σ-locally countable base;

(4) X is a countably-bi-quotient msss-image of a space having a σ-locally countable base.

Proof. (1) ⟹ (2) follows from Theorem 3.1 of [1] and Corollary 3.2 of [2].

(2) ⟹ (1). By Theorem 3.1, X is a sequential space with a σ-locally countable so-network P. It is easy to show that P is a σ-locally countable base for X.

(1) ⟹ (3) follows from Theorem 3.1 of [1].

(3) ⟹ (4) is obvious.

(4) ⟹ (1). Suppose X is the image of M under a countably-bi-quotient msssmap f, where M is a space with a σ-locally countable base. Because f is a msssmap, then there exists a base B for M such that P = {f(B) : B ∈ B} is a σ-locally countable network for X by Lemma 1.2 of [1]. Thus P is a σ-locally countable k-network for X by Lemma 2.5 of [1]. By Proposition 2.3.1 of [3], countably-bi-quotient maps preserve strong Fréchet property, thus X is a strong Fréchet space with a σ-locally countable k-network. Hence X has a σ-locally countable base by Theorem 3.9 of [6] and Proposition 3.2 of [13].

Corollary 3.3. A space with a σ-locally countable base is preserved by a countably-bi-quotient msss-map.

4. On sequence-covering msss-images

Theorem 4.1. The following are equivalent for a space X:

(1) X is a sequence-covering msss-image of a metric space;

(2) X is a sequentially quotient msss-image of a metric space;

(3) X has a σ-locally countable s-network.

Proof. (1) ⟹ (2) follows from Proposition 2.1.17 of [3].

(2) ⟹ (3). Suppose f : M → X is a sequentially quotient msss-map, where M is a metric space. Since f is a msss-map, there exists a base B for M such that P = {f(B) : B ∈ B} is a σ-locally countable network for X by Lemma 1.2 of [1]. Because s-networks are preserved by sequentially quotient maps by Lemma 3.1 of [4], X has a σ-locally countable s-network P.

(3) ⟹ (1). Suppose P is a σ-locally countable s-network for X. then P is a σ-locally countable cs^{∗}-network for X by Theorem 2.4 of [4]. Hence X is a sequence-covering msss-image of a metric space by Theorem 1 of [11].

The following corollaries can be proved by Theorem 4.1, Corollary 2.3 of [4], Lemma 3.1 of [4] and Proposition 2.1.16 of [3].

Corollary 4.2. A space X has a σ-locally countable sequential quasi-base if and only if X is a quotient msss-image of a metric space.

Corollary 4.3. A space X has a σ-locally countable Fréchet quasi-base if and only if X is a pseudo-open msss-image of a metric space.

5. On strong sequence-covering msss-images

Theorem 5.1. The following are equivalent for a space X:

(1) X is a strong sequence-covering msss-image of a metric space;

(2) X is a strong compact-covering msss-image of a metric space;

(3) X has a σ-locally-countable cs-network.

Proof. (1) ⟹ (3) follows from Lemma 1.2 of [1] and the fact: cs-networks are preserved by strong sequence-covering maps.

(3) ⟹ (2). Suppose P is a σ-locally-countable cs-network for X. Denote P = ⋃{P_{i} : i ∈ N}, where each P_{i} = {P_{α} : α ∈ A_{i}} is locally-countable in X. We can assume that each P_{i} is closed under finite intersections and X ∈ P_{i} ⊂ P_{i+1}. By the proof of Theorem 2.1, there exist a metric space M and a msss-map f : M → X. We will prove that f is a strong compact-covering map. For each sequence {x_{n}} converging to x_{0}, we can assume that all x′_{n}s are distinct, and that x_{n} ≠ x_{0} for each n ∈ N. Let K = {x_{m} : m ∈ ω}. Suppose V is an open neighborhood of K in X. A subfamily A of P is said to have the following property, which is denoted by F (K, V), if:

(a) A is finite,

(b) for each P ∈ A, ϕ ≠ P ∩ K ⊂ P ⊂ V,

(c) for each z ∈ K, there exists a unique P_{z} ∈ A such that z ∈ P_{z},

(d) if x_{0} ∈ P ∈ A, then K \ P is finite.

For each i ∈ N, put

P_{i}(K) = {A ⊂ P_{i} : A has the property F (K, X)};

then |P_{i}(K)| < ℵ_{0}. Denote P_{i}(K) by {P_{ij} : j ∈ N} (when P_{i}(K) = {P_{i1}, ⋯, P_{is}}, denote P_{ij} = P_{is} if j > s). For each n ∈ N, put

P′_{n} = ⋀_{i,j≤n} P_{ij},

then P′_{n} ⊂ P_{n} and P′_{n} also has the property F (K, X).

For each i ∈ N and each m ∈ ω, there exists α_{im} ∈ A_{i} such that x_{m} ∈ P_{α_{im}} ∈ P′_{i}. Let b_{m} = (α_{im}) ∈ ∏_{i∈N} A_{i}. It is easy to prove that {P_{α_{im}} : i ∈ N} is a network of x_{m} in X. Then b_{m} ∈ M and f(b_{m}) = x_{m} for each m ∈ ω. For each i ∈ N, there exists n(i) ∈ N such that α_{in} = α_{i0} when n ≥ n(i). Hence the sequence {α_{in}} converges α_{i0} in A_{i}. Thus, there is a sequence {b_{n}} converging to b_{0} in X. This shows that f is sequence-covering.

Since X has a σ-locally-countable cs-network, each compact subset L of X has a countable cs-network. So L is metrizable. We can prove that f is compactcovering by the proof of Theorem 2 in [5].

(2) ⟹ (1) is obvious.

Corollary 5.2. The following are equivalent for a space X:

(1) X is a strong sequence-covering and quotient msss-image of a metric space;

(2) X is a strong compact-covering and msss-image of a metric space;

(3) X is a k-space with a σ-locally-countable cs-network.

REFERENCES

[1] Shou Lin, Locally countable families, locally finite families and Alexandroff’s problem, Acta Math. Sinica, 37 (1994), 491–496 (in Chinese).

[2] Shou Lin, On sequence-covering s-mappings, Adv. in Math., 25 (1996), 548–551 (in Chinese).

[3] Shou Lin, Generalized Metric Spaces and Mappings, Chinese Scientific Publ., Beijing, 1995 (in Chinese).

[4] Shou Lin, Sequential networks and sequentially quotient images of metric spaces, Acta Math. Sinica, 42 (1999), 49–54 (in Chinese).

[5] Shou Lin, A note on Michael-Nagami’s problem, Ann. Math., 17 (1996), 9–12 (in Chinese).

[6] D. Burke, Closed mappings, In: Surveys in General Topology, New York, Academic press, 1980.

[7] J. Boone, F. Siwiec, Sequentially quotient mappings, Czech Math. J., 26 (1976), 174–182.

[8] F.Siwiec, On defining a space by a weak-base, Pacific J. Math., 52 (1974), 233–245.

[9] J. A. Guthrie, A characterization of ℵ_{0}-spaces, Gen. Top. Appl, 1 (1971), 105–110.

[10] S. P. Franklin, Spaces in which sequences suffice, Fund. Math., 57 (1965), 107–115.

[11] Kedian Li, Xiufeng Feng, Zhangshuai Liu, On msss-mappings, J. Math. Res. & Exp., 20 (2000), 223–226 (in Chinese).

[12] J. Nagata, General metric spaces I, In: Topics in General Topology, North-Holland, Amsterdam, 1989.

[13] G. Gruenhage, E. Michael, Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math., 113 (1984), 303–332.

[14] Shou Lin, Chuan Liu, Mumin Dai, Images of locally separable metric spaces, Acta Math. Sinica, New Series, 13 (1997), 1–8 (in Chinese).

[15] Y. Tanaka, Zhaowen Li, Certain covering-maps and k-networks, and related matters, Topology Proc., 27 (2003), 317–334.

[16] Zhaowen Li, A mapping theorem on ℵ-spaces, Mat. Vesnik, 57 (2005), 35–40.

[17] Zhaowen Li, Spaces with σ-locally countable weak-bases, Archivum Mathematicum, 42 (2006), 135–140.

[18] Zhaowen Li, Shouli Jiang, On msk-images of metric spaces, Georgian Math. J., 12 (2005), 515–524.

[19] Zhaowen Li, A note on ℵ-spaces and g-metrizable spaces, Czech. Math. J., 12 (2005), 803– 808.

(received 24.11.2005, in revised form 02.01.2007)

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P.R.China First author’s current address: Department of Information, Hunan Business College, Changsha, Hunan 410205, P.R.China